Obligated fasting in Christianity

A similar question What denominations don't believe in fasting? And Why? mentions the following under the heading Who fasts and who doesn't?:

• Both the Eastern Orthodox and Roman Catholic churches have required fasting periods for their members, as shown by the links.

• Members of the Church of Jesus Christ of Latter-Day Saints are encouraged to fast for the first Sunday of every month, which they call their fast and testimony meeting. Instead of speakers, members come up to the microphone and bear their testimony about the truthfulness of their experiences in the Gospel [per Gilbert LeBlanc]

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For a scriptural viewpoint, the Jehovah's Witnesses article What Does the Bible Say About Fasting? under the heading Is fasting required for Christians? supplies the following:

No. God required the Israelites to fast on Atonement Day, but he eliminated that observance after Jesus permanently atoned for the sins of repentant people. (Hebrews 9:​24-​26; 1 Peter 3:​18) Christians are not under the Mosaic Law, of which Atonement Day was a part. (Romans 10:4; Colossians 2:​13, 14) Therefore, each Christian may decide for himself whether to fast.​—Romans 14:​1-4.

Christians recognize that fasting is not the focus of their worship. The Bible never associates fasting with happiness. In contrast, true Christian worship is marked by joy, reflecting the personality of Jehovah, “the happy God.”​—1 Timothy 1:​11; Ecclesiastes 3:​12, 13; Galatians 5:​22.

Going by what Romans 14:​1-4 says, the decision would be up to the individual whether to fast or not. The article does give additional information on fasting that you may consider.

On a personal note, I have fasted both for dietary and spiritual reasons. As for the spiritual, I fasted during the anniversary of Christ's death to focus more on the spiritual than the physical.

Obligated fasting in Christianity?

Many denominations require their faithful to fast at various times of the year.

Fasting is a practice in several Christian denominations and is done both collectively during certain seasons of the liturgical calendar, or individually as a believer feels led by the Holy Spirit; many Christians also fast before receiving Holy Communion (this is known as the Eucharistic Fast). In Western Christianity, the Lenten fast is observed by many communicants of the Catholic Church, Lutheran Churches, Methodist Churches, Reformed Churches, Anglican Communion, and the Western Orthodox Churches and is a forty-day partial fast to commemorate the fast observed by Christ during his temptation in the desert. While some Western Christians observe the Lenten fast in its entirety, Ash Wednesday and Good Friday are nowadays emphasized by Western Christian denominations as the normative days of fasting within the Lenten season.

In the traditional Black Fast, the observant abstains from food for a whole day until the evening, and at sunset, traditionally breaks the fast. In India and Pakistan, many Christians continue to observe the Black Fast on Ash Wednesday and Good Friday, with some fasting in this manner throughout the whole season of Lent.

Partial fasting within the Ethiopian Orthodox Church (abstaining from meat and milk) which takes place during certain times of the year and lasts for weeks. - Fasting

What I find interesting about this question is the link the was mentioned in the question itself taken from Zechariah 8:19

19 Thus saith the Lord of hosts; The fast of the fourth month, and the fast of the fifth, and the fast of the seventh, and the fast of the tenth, shall be to the house of Judah joy and gladness, and cheerful feasts; therefore love the truth and peace.

This reminds me of the fast of Ember Days also commonly known as the ”fasts of the four seasons”. Although most Catholics now do not participate in this traditional fast, it is still practiced by a few Catholics here and there. When Pope Paul VI amended the Catholic Liturgical practices, he left the Ember Day fast open to the discretions of the Episcopal National Conference of Bishops if they wanted to keep this tradition.

In the liturgical calendar of the Western Christian churches, ember days are four separate sets of three days within the same week—specifically, the Wednesday, Friday, and Saturday—roughly equidistant in the circuit of the year, that are set aside for fasting and prayer. These days set apart for special prayer and fasting were considered especially suitable for the ordination of clergy. The Ember Days are known in Latin as the quatuor anni tempora (the "four seasons of the year"), or formerly as the jejunia quatuor temporum ("fasts of the four seasons").

The four quarterly periods during which the ember days fall are called the embertides.

The term Ember days refers to three days set apart for fasting, abstinence, and prayer during each of the four seasons of the year. The purpose of their introduction was to thank God for the gifts of nature, to teach men to make use of them in moderation, and to assist the needy.

Possibly occasioned by the agricultural feasts of ancient Rome, they came to be observed by Christians for the sanctification of the different seasons of the year. James G. Sabak argues that the Embertide vigils were "...not based on imitating agrarian models of pre-Christian Roman practices, but rather on an eschatological rendering of the year punctuated by the solstices and equinoxes, and thus underscores the eschatological significance of all liturgical vigils in the city of Rome."

At first, the Church in Rome had fasts in June, September, and December. The Liber Pontificalis ascribes to Pope Callixtus I (217–222) a law regulating the fast, although Leo the Great (440–461) considers it an Apostolic institution. When the fourth season was added cannot be ascertained, but Pope Gelasius I (492–496) speaks of all four. The earliest mention of four seasonal fasts is known from the writings of Philastrius, bishop of Brescia (died ca 387) (De haeres. 119). He also connects them with the great Christian festivals.

As the Ember Days came to be associated with great feast days, they later lost their connection to agriculture and came to be regarded solely as days of penitence and prayer. It is only the Michaelmas Embertide, which falls around the autumn harvest, that retains any connection to the original purpose.

The Christian observance of the seasonal Ember days had its origin as an ecclesiastical ordinance in Rome and spread from there to the rest of the Western Church. They were known as the jejunium vernum, aestivum, autumnale and hiemale, so that to quote Pope Leo's words (A.D. 440 - 461) the law of abstinence might apply to every season of the year. In Leo's time, Wednesday, Friday and Saturday were already days of special observance. In order to tie them to the fasts preparatory to the three great festivals of Christmas, Easter and Pentecost, a fourth needed to be added "for the sake of symmetry" as the Encyclopædia Britannica 1911 has it.

From Rome the Ember days gradually spread unevenly through the whole of Western Christendom. In Gaul they do not seem to have been generally recognized much before the 8th century.

Their observance in Britain, however, was embraced earlier than in Gaul or Spain, and Christian sources connect the Ember Days observance with Augustine of Canterbury, AD. 597, said to be acting under the direct authority of Pope Gregory the Great. The precise dates appears to have varied considerably however, and in some cases, quite significantly, the Ember Weeks lost their connection with the Christian festivals altogether. Spain adopted them with the Roman rite in the eleventh century. Charles Borromeo introduced them into Milan in the sixteenth century.

In the Eastern Orthodox Church, ember days have never been observed. Yet in Western Rite Orthodoxy, which is in full communion with the Eastern Orthodox, the Ember days are observed. Ember Days (Wikipedia)

For those Catholics pondering a small way in keeping with the spirit of this tradition, without fasting may I recommend having a taste of the past in the form of a Medieval Ember Day Tart at dinner.

Ember Day Tart

Should we as Christians fast? This question I will not get into here, but I will simply let St. Matthew have the last words on the issue from his Gospel.

14 Then came to him the disciples of John, saying, Why do we and the Pharisees fast oft, but thy disciples fast not?

15 And Jesus said unto them, Can the children of the bridechamber mourn, as long as the bridegroom is with them? but the days will come, when the bridegroom shall be taken from them, and then shall they fast. - Matthew 9: 14-15

What is more is that Christ himself showed us the example of fasting while in the desert. This was done just prior to his public ministry and Is commonly associated with the Temptation of Christ.