Fasting days
Whether days of fasting are kept or whether fasting is free, as the two passages seem to weigh.
Romans 14:1-6
1Now accept one who is weak in faith, but not for disputes over opinions.
2One man has faith to eat all things, but he who is weak eats only vegetables.
3Don’t let him who eats despise him who doesn’t eat. Don’t let him who doesn’t eat judge him who eats, for God has accepted him.
4Who are you who judge another’s servant? To his own lord he stands or falls. Yes, he will be made to stand, for God has power to make him stand.
5One man esteems one day as more important. Another esteems every day alike. Let each man be fully assured in his own mind.
6He who observes the day, observes it to the Lord; and he who does not observe the day, to the Lord he does not observe it. He who eats, eats to the Lord, for he gives God thanks. He who doesn’t eat, to the Lord he doesn’t eat, and gives God thanks.
Matthew 9:14-17
14Then John’s disciples came to him, saying, “Why do we and the Pharisees fast often, but your disciples don’t fast?”
15Jesus said to them, “Can the friends of the bridegroom mourn, as long as the bridegroom is with them? But the days will come when the bridegroom will be taken away from them, and then they will fast.
16No one puts a piece of unshrunk cloth on an old garment; for the patch would tear away from the garment, and a worse hole is made.
17Neither do people put new wine into old wine skins, or else the skins would burst, and the wine be spilled, and the skins ruined. No, they put new wine into fresh wine skins, and both are preserved.”
What each tradition says
Rabbinic / Midrashic
The source states that the Day of Atonement is the only fast-day prescribed by the Mosaic law. After the Captivity, four regular fast-days were established to commemorate various sad events that had befallen the nation during that period, as referenced in Zechariah.
Julius H. Greenstone, Emil G. Hirsch, Hartwig Hirschfeld (Jewish Encyclopedia editorial board), The Jewish Encyclopedia (1905), article "Fasting and Fast-Days" — List of Fast-Days
Patristic
Augustine says that where the divine Scriptures have laid down no definite rule, the custom of the people of God or practices instituted by their fathers are to be held as the law of the Church. Thus, in such undefined matters, church custom and ancestral practice carry the authority of law.
Augustine of Hippo, Letters of St. Augustin, Letter XXXVI (A.D. 396) to Casulanus (NPNF series 1, vol. 1, ed. Schaff) — Letter XXXVI, Ch. I, sec. 2 (on fasting on the seventh day)
Catholic
Aquinas holds that fasting in general is a matter of precept of the natural law. The fixing of the time and manner of fasting, as becoming and profitable to the Christian people, is a matter of precept of positive law established by ecclesiastical authority. The latter is the Church fast; the former is the fast prescribed by nature.
Thomas Aquinas, Summa Theologiae, Second Part of the Second Part, Q. 147 (Second and Revised Edition, 1920, trans. Fathers of the English Dominican Province) — Q. 147, Art. 3, "Whether fasting is a matter of precept?", I answer that
Orthodox
St. Philaret teaches that, as the Old Testament Church included other appointed festivals and fasts under the name Sabbath, so the Christian Church is now bound to keep certain other days besides the Lord's Day. These are appointed as festivals to God's glory and the honor of the Blessed Virgin and other saints, or as days of fasting.
St. Philaret (Drozdov) of Moscow, The Longer Catechism of The Orthodox, Catholic, Eastern Church (Examined and Approved by the Most Holy Governing Synod, Moscow 1830) — Part III, On the Fourth Commandment, Q. 541
Reformed / Protestant
Calvin says we should not place holiness in outward and indifferent matters, but restrain ourselves with moderation and equity, not restricting others to what we approve and allowing each to keep his freedom. Regarding fasting and prayers, John gave his disciples a particular training, with stated days for fastings and fixed hours of prayer.
John Calvin, Commentary on Matthew 9:14 (Harmony of the Evangelists, vol. 1, Calvin's Commentaries) — Comment on Matthew 9:14 (section on Matthew 9:14-17)
Evangelical
Wesley explains that while Jesus was with his disciples, it was a time of rejoicing, not mourning, since fasting and mourning go together. After Jesus was gone, Wesley says, all his disciples would likewise be in fastings often.
John Wesley, John Wesley's Notes on the Bible, Notes on the Gospel according to St Matthew — Note on Matthew 9:15