Can you be Christian if you don't believe in God?

This is a rather interesting question...

It depends on what you mean by Christian. I generally label Christians as professors of the Nicene Creed (or a similar creed). The creed is very long, and I don't wish to type it all out here. If you click on the hyperlink, it will show you what the creed says. Let me instead give a quick (and admittedly incomplete, though that isn't very relevant for this question) version of the summary.

We believe in the existence of the Trinity, which includes a God, the same essence, in three separate parts, including the Father, Son, and Holy Spirit. We believe in the Church that he established, and that he is coming back to judge the living and dead one day.

Those are the most relevant parts of the creed, and I did skip over certain parts of what "The Church" and other things are, but let us continue as an atheist probably wouldn't care.

Well we have already reached a problem! The existence of a Triune God is the literal first part of the creed.

So, for a short answer to your question, no. An atheist or other non-believer type cannot be a Christian.

However there are some beliefs with nuances that are related, such as Christian Fideism, the view that there is no evidence for the existence of God, but we still ought to believe in him. These are certainly Christian people. There are also people who would adopt a Christian moral standard (with some exceptions), that could fall under a category of "being" a Christian.

I just realized that I didn't fully answer your question! Allow me to bring up what the Bible specifically says on this matter.

While the Bible never really says what makes someone a "Christian" (mainly since this term was never used by original Christians and instead was given to them by hostile outsiders). Instead, the Bible says what a "saved" person is. the Bible clearly states that if you believe in God and Jesus and in your heart believe that God the Father raised Jesus from the dead, you will be saved (John 3:16). This is the minimum requirement for salvation.

So my answer is still the same, but that more directly answers your question!

No - to be a "Christian" demands one 'follow' the person called Christ. Follow - trust, believe, respect, honour, etc. By believe, we must accept all he said and taught.

We don't really get to pick and choose what we believe based on our inner wisdom or understanding. We have to accept everything he said without question or even doubt. Easier said than done, but that is the expectation.

He calls us to worship the same God he had, so to not believe in God is to simply pretend that you are a Christian for your own personal theories and live in a deluded fantasy. The world is full of religions and sadly, Christianity has many denominations which differ wildly from each other and fail in various ways to truthfully represent the Jesus they claim to follow - what they all variably follow is a man-made god which they attract others to - all men need to follow something or aspire to something grander than themselves. The more creatively they construct the mystery, the more they will attract. Christianity is not a mystery - but it is a secret!

The whole point on being a Christian is to obtain the next life. This was a promise of Jesus he reiterated from his Father and God. To pretend one is a Christian will not bring any of the benefits of being one - again, this would simply be delusional.

Finally, the whole point of being a Christian is not about the Christian - it is about God. You are essentially elevating 'Christianity' above God!

Is there anything in Christian canon that stipulates a belief in God is required to call oneself a Christian?

Well, if you want to get pedantic, there is nothing stopping you from claiming to be "Christian" regardless of your beliefs. Indeed, this very topic can be hotly contested on this very site with respect to LDSs and JWs.

To actually be Christian means, literally, to follow the teachings of Christ. If I were to trot out everything Christ is recorded to have said, there would be plenty of instances where following Christ's teachings would be... challenging (to say the least) for someone that disbelieves in God. (On that note, see also this question​.)

That said, one might summarize the Christian message as follows:

• God Created the world, humans, and you. This Creation was Perfect ("Very Good").
• Humans (Adam and Eve) broke God's command; that is, they sinned.
• God sent his perfect Son to atone for sin... yours, mine, and everyone's, everywhere, everywhen.
• That atonement was accomplished by Christ's death and resurrection.

However, I also want to look at something else you mentioned:

A reason an atheist would want to participate in Christian social frameworks [...] is if they compartmentalize the foundational dogma/mythology, which they find false or contrary to evidende (sic)

I surmise that here you are talking about "science", and particularly the outgrowths of philosophical naturalism (PN) which are Uniformitarianism and Evolutionism (collectively, U&E). Now, PN on its own necessarily denies God's existence; indeed, doing so is in many senses the very definition of PN. Accordingly, while many people may believe in God and Christ while simultaneously accepting the "accepted science" of U&E, these philosophies are fundamentally in contradiction.

Now, it should be obvious that Christ's Purpose is rendered void if there is no Sin. It should be at least somewhat obvious that the notion of Sin makes no sense without Law, and there can be no Law if there is no God. Thus we have our first serious objection to your premise.

What's less obvious is that the entire concept of sin makes no sense if the world was not Very Good prior to the Fall (hence the first point). The important point here is that a Very Good Creation is not compatible with billions of years of death and suffering (i.e. U&E; further reading). Worse, what too many Christians don't realize — because Satan and his anti-Christian minions have worked long and hard to bury this understanding — is that "the evidence" is NOT contradictory to Christianity, including a plain reading of Genesis that includes Creation in ~144 hours (as measured on Earth) approximately 6-10 kya. Rather, there are interpretations of evidence, made from an anti-Christian perspective, which purport — dishonestly, maliciously and falsely — to repudiate the history of the Bible, which comes from God and is reaffirmed by Christ in Mark 10:6.

Moreover... Christ certainly believed in God. (Christ is God, according to Trinitarians, but that's a whole other topic.) Therefore, one who believes in Christ ought also to believe in God. One who believes in God has no need to resort to the extreme lengths UNs to explain Creation. Interpretations of available scientific evidence which are also consistent with biblical history are well known, well studied, and at least as consistent as U&E interpretations.

Accordingly, I must additionally question your supposed need for a "Christian" that does not believe in God. I would commend you (and every other Christian) to start from Christ and seriously examine any competing beliefs (U&E are beliefs; they are not "facts", and they have numerous and significant problems) and whether or not they are really "false or contrary to evidence". Don't just take the PNs word for it; their philosophy is fundamentally opposed to Christ, and vice versa (n.b. 1 Corinthians 1:23, and also 2 Thessalonians 2:10-11).

That all said, I'm tempted to be contrarian and say "sure, why not!". Humans are surprisingly good at doublethink. Plenty of Christians accept Christ's message and the anti-Christian messages of PN/U&E. It may be easier in that case because the contradiction is perhaps less obvious, yet if many people can do that, why not simultaneously believe in Christ and disbelieve in God? I would venture even to say that most Christians (and yes, that includes myself) believe in something that is contrary to God's Word.

If you're doing so consciously and deliberately, that's probably an issue, but I am also not the Judge of Man (and am trying to be mindful of Matthew 7:3-5).

This question could generate a lot of different answers, as each denomination has its own definition of what it means to be a Christian.

Given two denominations, each could reasonably and with certainty say that they are true Christians and that the other church is Christian in name only.

Which one is right? It wouldn't be difficult to find groups that would answer "Neither! They are both wrong! We are the true Christians!".

But what really makes this a difficult question to answer is the many denominations that call themselves Christian but barely have any concept of a personal God. God is a feeling within you. Jesus said so. Love, peace, sharing, helping, etc. are what they preach and practice.

For a real example of this last case, see my answer to Do any Christian groups or denominations not see having a definite doctrinal position on God's nature as essential for salvation? - Christianity Stack Exchange.

A Christian is someone who is a follower or disciple of Jesus Christ. Jesus taught repeatedly and firmly about God, so it would be very difficult to be one of his disciples if you reject his teachings, and even arguably his foundational teachings. When asked what the greatest commandment is, Jesus replied that it is to "love the Lord your God with all your heart, and with all your soul, and with all your mind." Those who deny the existence of God neither follow Jesus's teachings, nor do they position themselves as followers of him who are willing to be conformed to him, instead they sit above him, overruling his teachings and declaring him to be wrong.

So-called Christian atheists are a thing, but I don't know of any other Christians who would consider them to actually be Christians. The existence of God is too foundational to Christianity to be denied.

No.

He that cometh from above is above all: he that is of the earth is earthly, and speaketh of the earth: he that cometh from heaven is above all. And what he hath seen and heard, that he testifieth; and no man receiveth his testimony. He that hath received his testimony hath set to his seal that God is true. - John 3:31-33

It is not possible to receive the testimony of the Son of God and yet not possess a belief in God.