Is there a Buddhist equivalent to Lutheran doctrine of Justification?

Yes and no.

No: The term “sin” does not have any special connotation in Buddhism, as it has in major theistic religions like Christianity, Judaism, or Islam. In all these religions, the general belief is that sins are individual actions which are contrary to the will of God or to the will of the Supreme Being. The karmic principle is based on action and consequence.

In Buddhist thinking the whole universe, men as well as gods, are subject to a reign of law. Every action, good or bad, has an inevitable and automatic effect in a long chain of causes, an effect which is independent of the will of any deity. Even though this may leave no room for the concept of 'sin' in the sense of an act of defiance against the authority of a personal god, Buddhists speak of 'sin' when referring to transgressions against the universal moral code. In addition, Buddhism has no concept of "original sin" like Christianity.

Yes: If you consider all of the above:

• deities are subject to the same karmic principles as all other living beings
• there is no comparable concept of sin, just action and consequence

then the principle of Pratītyasamutpāda comes very close. Commonly translated as dependent origination, or dependent arising, it is a key doctrine of Buddhist philosophy, which states that all dharmas arise in dependence upon other dharmas, i.e. that everything is interconnected and nothing can truly be separate.

It may be that you've misunderstood the Christian sola fide doctrine -- I think that at least some Christian sects would hold that relatively blameless behaviour is evidence of salvation or grace. Still, perhaps you're right -- the table in that Wikipedia article says that the (specifically) Lutheran belief is ...

Justification is separate from and occurs prior to sanctification

... implying that they're not always concurrent.

Incidentally I haven't read your quote from Luther in context -- but from what I remember of my "history of religion" class, Luther was writing in reaction to a time when the church was selling indulgences:

One particularly well-known Catholic method of exploitation in the Middle Ages was the practice of selling indulgences, a monetary payment of penalty which, supposedly, absolved one of past sins and/or released one from purgatory after death. It was the selling of indulgences that led the Reformer Martin Luther to post his famous 95 Theses - a document challenging Roman Catholic authority in theological matters, including indulgences and many others. Luther's opposition to the selling of indulgences was not new, however. In most of the Reformation movements stress lay not upon new understandings or doctrines, but on a return to the more authentic and original excellence of tradition.

Luther, one of the main Protestant Reformers, eventually arrived at the conclusion that divine relationship and salvation come by grace through faith, not by good works, belief in dogma, or economic propitiation. One's relationship to the divine is initiated by God, and one can only participate in this relationship by remaining open to it. Therefore, Luther's theology placed him in square opposition to the Roman Catholic practice of selling indulgences.

Perhaps we could all agree or not disagree with Luther's opinion about that -- and even try to understand Luther's claim in that context.

You might want to read this topic -- What are the differences/similarities in the concept of faith as used in Buddhism and Christianity?

MN 125 might be ambiguous on the topic (not that the sutta is ambiguous but that I'm not sure how to apply it to your question about Lutheranism).

One the one hand it says ...

Prince Jayasena dwells in the midst of sensual pleasures, enjoying them, consumed by thoughts of them, burning with fever for them, and eagerly seeking more. It’s simply impossible for him to know or see or realize what can only be known, seen, and realized by renunciation.

... which implies they're connected (can't have faith or vision, right view, without right conduct).

On the other hand it says ...

A householder hears that teaching, or a householder’s child, or someone reborn in some clan. They gain faith in the Realized One, and reflect, ‘Living in a house is cramped and dirty, but the life of one gone forth is wide open. It’s not easy for someone living at home to lead the spiritual life utterly full and pure, like a polished shell. Why don’t I shave off my hair and beard, dress in ocher robes, and go forth from the lay life to homelessness?’

After some time they give up a large or small fortune, and a large or small family circle. They shave off hair and beard, dress in ocher robes, and go forth from the lay life to homelessness. And it’s only then that a noble disciple comes out into the open, for gods and humans cling to the five kinds of sensual stimulation.

... which implies one happens before the other.

So far as I know, Theravada might tend to put the two together -- I tend to associate stream-winning with faith and virtue -- for example 55.27:

When an educated noble disciple has four things, they’re not frightened or terrified, and don’t fear what awaits them after death. What four? Firstly, a noble disciple has experiential confidence in the Buddha … Seeing in themselves that experiential confidence in the Buddha, they’re not frightened or terrified, and don’t fear what awaits them after death.

Furthermore, a noble disciple has experiential confidence in the teaching …

Furthermore, a noble disciple has experiential confidence in the Saṅgha …

Furthermore, a noble disciple’s ethical conduct is loved by the noble ones, unbroken, impeccable, spotless, and unmarred, liberating, praised by sensible people, not mistaken, and leading to immersion. Seeing in themselves that ethical conduct loved by the noble ones, they’re not frightened or terrified, and don’t fear what awaits them after death.

When an educated noble disciple has these four things, they’re not frightened or terrified, and don’t fear what awaits them after death.”

“Sir, Ānanda, I am not afraid. What have I to fear? For I have experiential confidence in the Buddha … the teaching … the Saṅgha … And of the training rules appropriate for laypeople taught by the Buddha, I don’t see any that I have broken.”

“You’re fortunate, householder, so very fortunate, You have declared the fruit of stream-entry.”

Admittedly "experiential confidence" there is aveccappasāda which is arguably more than only "faith".

Also I think I'm told that even stream-winners are still subject to some anger, conceit (e.g. associated with disputes), desires -- that seems to me a limit to their "virtue" even if they are a stream-winner.

The Śrāvaka article mentions:

The canon occasionally references the "four pairs" and "eight types" of disciples. This refers to disciples who have achieved one of the four stages of enlightenment:

• Sotāpanna
• Sakadāgāmin
• Anāgāmin
• Arahat

In regards to disciples achieving arahantship, Bhikkhu Bodhi writes:

In principle the entire practice of the Noble Eightfold Path is open to people from any mode of life, monastic or lay, and the Buddha confirms that many among his lay followers were accomplished in the Dhamma and had attained the first three of the four stages of awakening, up to nonreturning (anāgāmi; Theravāda commentators say that lay followers can also attain the fourth stage, arahantship, but they do so either on the verge of death or after attainment immediately seek the going forth [that is, homelessness, associated with becoming a monastic]).

For each of these stages, there is a "pair" of possible disciples: one who is on the stage's path (Pāli: magga); the other who has achieved its fruit (Pāli: phala). Thus, each stage represents a "pair" of individuals: the path traveler (Pāli: maggattha) and the fruit achiever (Pāli: phalattha). Hence, the community of disciples is said to be composed of four pairs or eight types of individuals (Pāli: cattāri purisayugāni attha purisapuggalā).

That implies there are people on the path who haven't yet attained the fruit -- perhaps they "have faith but haven't yet attained virtue and the fruit of virtue" -- which again might be compatible with the Lutheran position.

I think some Mahayana texts mention "hearers" -- people who have heard the doctrine.

And maybe "followers" -- people who follow the doctrine -- is a separate and more elevated or praise-worthy category.

So again that maybe like the Lutheran.

I find the answer to this topic insightful -- What is meant by Namu Amida Butsu and also Nam-myoho-renge-kyo? -- explaining how "faith" is connected with "realisation".

Personally I think that the grossest meaning of the sola fide doctrine is that good deeds isn't enough -- i.e. that faith is necessary as well -- "faith as well" and not necessarily "faith alone" or "faith instead".

That has a parallel in Buddhist doctrine, e.g. SN 42.6 which says that good deeds -- or more especially "rites and rituals" -- can't make up for the case where ...

the case where a man is one who takes life, steals, indulges in illicit sex; is a liar, one who speaks divisive speech, harsh speech, & idle chatter; is greedy, bears thoughts of ill-will, & holds to wrong views

I think there's a similar message in the original Christian doctrine (by which I mean the Gospels) -- e.g. discouraging public prayer, praising the widow's mite, and probably more.

Buddhism has a notion of "merit" -- e.g. "making merit". You might see that as a precursor to enlightenment. Some might even see it as a wrong goal, where or because it's pursued instead of enlightenment.

I presume that a Buddhist's "meritorious" deeds (and attempting to keep the precepts is meritorious IMO) are often the result of "faith" -- and in that sense not unlike the Lutheran doctrine of salvation after justification (although, to be clear, there's doctrine saying that merit and ethics are intermediate not ultimate goals).

Incidentally I found the source of the first quoted sentence. It's from here -- Let Your Sins Be Strong: A Letter From Luther to Melanchthon -- the last paragraph:

If you are a preacher of mercy, do not preach an imaginary but the true mercy. If the mercy is true, you must therefore bear the true, not an imaginary sin. God does not save those who are only imaginary sinners. Be a sinner, and let your sins be strong, but let your trust in Christ be stronger, and rejoice in Christ who is the victor over sin, death, and the world. We will commit sins while we are here, for this life is not a place where justice resides. We, however, says Peter (2. Peter 3:13) are looking forward to a new heaven and a new earth where justice will reign. It suffices that through God's glory we have recognized the Lamb who takes away the sin of the world. No sin can separate us from Him, even if we were to kill or commit adultery thousands of times each day. Do you think such an exalted Lamb paid merely a small price with a meager sacrifice for our sins? Pray hard for you are quite a sinner.

There's a lot of commentary on it on the net (e.g. here -- Did Luther really tell us to ‘love God and sin boldly’).

As I understand it the message is, "Ever man is a sinner, and it does no good to pretend otherwise. Nothing you can do makes you less of a sinner, but even a great sinner should have a strong trust in Christ and Christ's victory."

Incidentally there may be some play of words in there that I don't understand, about there being no "justice" in this world:

We will commit sins while we are here, for this life is not a place where justice resides

... which I guess might be related to the doctrine's talking about being "justified" by faith (and possibly in the afterlife at that).

But now to answer your question, it seems to be that does have "Buddhist equivalent":

• In the description of Shin that I referenced previously:

  For such a person, calling Namu-Amida-Butsu is a willingness to rely on the Other-Power of Amida (on his practice of saving sentient beings)

  "Namu-Amida-Butsu" is then understood entirely differently: not as something we do, not as our own practice, but as an embodiment of Amida's practice. Another way to put it, once one totally abandons the idea that it's possible to "save yourself" through enough effort, one totally relies on Other-Power and so there is nothing to practice anymore on our part. There's just an absolute passivity of "deep listening to the Dharma". One realizes [etc.]

• In taking refuge -- perhaps not the ceremony of it ("doing"), but as an act of faith

• In the story of Aṅgulimāla -- which implies that taking refuge in the three jewels is possible and effective even for the great sinner.

I notice that there's even something akin to "rebirth" in the Aṅgulimāla Sutta:

The Buddha said to him, “Well then, Aṅgulimāla, go to that woman and say this:

‘Ever since I was born, sister, I don’t recall having intentionally taken the life of a living creature. By this truth, may both you and your baby be safe.’”

“But sir, wouldn’t that be telling a deliberate lie? For I have intentionally killed many living creatures.”

“In that case, Aṅgulimāla, go to that woman and say this:

‘Ever since I was born in the noble birth, sister, I don’t recall having intentionally taken the life of a living creature. By this truth, may both you and your baby be safe.’”

That might be veering off-topic -- the question seemed to be more about sinning than about the afterlife, and perhaps this is even a thicket of views according to Buddhist doctrine -- but perhaps that's not unlike, anyway I find it reminiscent of, passages in the Christian Bible like Romans 6.