At first blush, Romans 14:5-6 seems to say that it makes no difference to God which days we keep holy. Actually, these verses do not concern any days that must be kept holy. This is proved by the context of the entire chapter.
Paul admonishes the saints at Rome to receive the "weak in the faith" and not to sit in judgment of them (verse 1). Some of those recently converted, not yet having grown strong in the faith, refused to eat meat and subsisted mainly on vegetables.
The apostle explains why in another of his letters. Most of the available meat in the city had been offered to idols. Some Gentiles who had been converted and come out of idolatry still held some superstitious beliefs. They thought that idols actually had power over their lives. Therefore, "some, with consciousness of the idols," ate meat "as a thing offered to an idol" (I Corinthians 8:7). Paul assures them that "we know that an idol is nothing in the world, and that there is no other God but one" (verse 4).
But why does Paul break into his discussion about eating or refraining from eating meat to mention "esteem[ing] a day"? Notice the answer from within these very verses:
One person esteems one day above another; another esteems every day alike. Let each be fully convinced in his own mind. He 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; and he who does not eat, to the Lord he does not eat, and gives God thanks. (Romans 14:5-6)
Not only were some weak converts afraid of eating meat offered to idols, but others customarily abstained from a particular food—they practiced a kind of fast—on certain days, much as devout Catholics abstain from meat on Friday. Others regarded all days alike as far as eating was concerned.
The whole matter involves abstention from foods on particular days. "To eat or not to eat" is the question at hand. Paul is not referring to God's Sabbath or holy days at all!
Jesus says that we should fast before God and not be seen or let it be known by others unnecessarily (Matthew 6:16). But Jews and Gentiles both practiced semi-fasts on particular days of each week or month. Though divided on the matter, the Jews customarily fasted "twice in the week" (Luke 18:12) and on specific days of certain months (Zechariah 7:4-7). The Gentiles also were of various opinions over when to abstain from certain foods. These things are mentioned in Hasting's Encyclopedia of Religion and Ethics.
In God's sight, it does not matter when one abstains or fasts, but He does care whether we do it in a proper attitude and for the right reasons (see Isaiah 58). Paul wants the brethren to live at peace with one another and not argue or judge each other over their human opinions, which he calls "doubtful things" (Romans 14:1).
The Bible elsewhere teaches very plainly which days God made holy and commands us to keep holy. They are found in Exodus 20:8-11 and Leviticus 23.
The Fourth Commandment (Part One) (1997)
The Fourth Commandment (Part Two): Christ's Attitude Toward the Sabbath
The Fourth Commandment
Abstinence - Asceticism or Christian Living?
Are the Sabbath and Holy Days Done Away?
Does Paul Condemn Observing God's Holy Days?
Was Jesus Resurrected on Easter Sunday?
Rome's Challenge (Part 1)
Holy Days: God's Plan in the Holy Days
How Do We Keep God's Festivals?