The main issue is whether the Jewish Sabbath, as revealed to us in the Old Covenant, continues in the New Covenant age or whether Christians should meet on Sunday. Christians believe that as the Old Covenant faded away into the glorious revelation of the New, the Old Covenant Sabbath observation discontinued out of necessity.
For if we want to hold to the Jewish Sabbath then we would have to say, if we are consistent, that all the laws of the Old Covenant are still in effect now (including the priesthood, sacrifices, punishments etc.). This is why in Colossians 2:16 Paul can say we are not to judge one another in food or drink, festivals or Sabbaths: the old is passing away.
We have to wrestle with this text to understand its true meaning but it seems obvious that the Sabbath as commonly practiced by the Jews has no hold on Christians now. Note that the context says that Christ has “wiped out the handwriting of requirements that was against us.” The meaning of “wiped out” is to destroy, or make clean. And then Paul says “so” in verse 16 to connect it to the meaning of verse 14 and 15. In other words he says (my paraphrase) ‘since Christ in his atonement destroyed the necessity of keeping these laws, you no longer should judge each other in these matters.’
Added to this is the evidence of the meaning of the word ‘Sabbath’ for it does not mean ‘Saturday’ but simply means ‘rest’ or ‘stop.’ Therefore, the fact that the Jews rested or stopped working on Saturday is a particular application of the fourth commandment but not of its essence. “Remember the Sabbath day” is to do our work in six days and rest the seventh (Exodus 20:8-10). It doesn’t specify in the text what day of the week the Sabbath is. For a Christian to work Monday-Saturday and then to rest on Sunday in no way violates the fourth commandment because they are living by it’s basic principle. And indeed the principle is the thing: the controversy is not over the commandment per se, but its application. In other words the moral idea is the same ‘rest and worship’ but the application, the day, has changed. But why?
If we want to get to the meaning or application of the Sabbath in the New Testament we must see it in its fullness; it’s relation to the Lord of the Sabbath (Mark 2:28). This is evident from 1 Corinthians 16:1-2. In context we are told to abound “in the work of the Lord, knowing that [our] labor is not in vain in the Lord.” (1 Corinthians 15:58) This is why we gather to give offerings on “the first day of every week” (16:2), which are a token of our thankfulness to God in his deliverance of us from death and sin (1 Corinthians 15:56-57). Note that this is similar to the Old Testament practice of bringing animals and birds to be sacrificed on the Sabbath. (Numbers 28:9-10) Offerings in the Old Testament were a sign of forgiveness as well as thankfulness, just as they are in the New Testament.
The author of Hebrews reflects on this fulfillment aspect of the Sabbath as well when he says that “a rest for the people of God” remains (Hebrews 4:9) The rest in this passage reflects the rest which we receive from God by faith in His Son Jesus Christ. The place of rest is heaven because the readers are warned in verse 6 of chapter 4 that some did not enter into it. For there we “have a great High Priest,” Jesus Christ. (4:14). He has entered “into heaven itself” because he has offered Himself to God for our sins (Hebrews 9:24-26) The Sabbath day in the New Testament, therefore, must specifically reference Christ’s atoning work. The Messiah was resurrected; He is now seated in heaven. He then imparts life through His atonement as the resurrected Lord because His work is finished. This life is imparted to us by the Holy Spirit and results in our regeneration unto eternal life (1 Corinthians 15:51-57). And the day of resurrection, the day of life, was on the first day of the week or Sunday, as we call it (Luke 24:1,6) In other words, the New Testament is relating to us the observance of the Sabbath through the lens of the new redemptive situation.
As one scholar has noted, Peter speaks in a similar vein in Acts 4:10-11. He is speaking about the risen Lord. (vs. 10) Verse 11 is a quotation from Psalm 118:22 to prove that He still lives or reigns in power insofar as the cripple is healed in His name. In the context of the quote from Psalm 118 we read “this is the day the Lord has made; we will rejoice and be glad in it.” Again, the resurrection brings forth a new perspective; one wherein a man who could not walk now can (the power of the resurrection). This is related to the day of resurrection so as to make us rejoice and be glad! Thus in the worship of the saints we commemorate the resurrection of Christ as the new Sabbath day which was made.
We should also note that in Acts 20:7 Paul gathered with the disciples (“on the first day of the week”), which should be understood to be as Sunday. (cf. 1 Corinthians 16:2). Why? Because verse six mentions “for seven days” which is an indicator that the week has passed since in Jewish practice Friday-Saturday was a week, so the Saturday functions as the seventh day. The next day, “the first day” is a day where the disciples gather, break bread and hear Paul preach. Thus Sunday is the acceptable day of worship for the New Covenant people.
We must remember this is the pattern in the Old Testament: a continual cycle of seven days. The eighth day (the day after the Sabbath) has a rich meaning behind it as (see Leviticus 23:36,39; Numbers 29:35) reflected in the New Testament fulfillment in Christ. On the eighth day, in certain circumstances, there was to be no work but there was atonement offered. So Christ rested on the resurrection day from His works in relation to His atonement (cf. discussion above on Hebrews 4).
Furthermore, Christ is called the “first-fruits” of the resurrection in 1 Corinthians 15:20&23 which is a reference back to the Old Testament practice of offering the first part of the harvest unto God. This was given on the day after the Sabbath (Leviticus 23:10-11), that is, the eighth day. Thus our Sunday worship is a commemoration of the New Testament first-fruit, Jesus Christ, whose resurrection is now a sure promise of our resurrection.
By now it should be clear by what I have said that Sunday observance is not a matter of finding a text that says, “worship on Sunday.” It is by inference that we establish this truth. As Reformed Christians we do believe in a type of Sabbath observance. For the fourth commandment is not based on theocratic law (laws that pertained only to the Jews in the state of Israel) but creation law. After all, God reminds even the Jews of their obligation to keep the Sabbath for the very reason that God Himself rested on the seventh day of creation. (Exodus 20:8-11) There are differences in the way the Sabbath is upheld between the Old and the New Testament but the essence of the commandment remains.
In conclusion we cannot just ignore the fourth commandment; it is applicable today as ever. But it does not command us now to keep the Jewish ceremonies. Indeed all of the Ten Commandments are relevant for us today but not in the exactly the same way that they were in the Old Testament. The ceremonies of the Old Covenant are done away in Christ, but the essence of the Law remains. If we obey the fourth commandment we must observe it unto the Lord Jesus Christ but we cannot be bound to Saturday but to Sunday: the new Sabbath day that He has made.
 Just as Christians believe in the seventh commandment and yet have ‘tightened’ the restrictions in accordance with Jesus’ design and teaching (for example, with particular application to the issue of divorce – see Matthew 19:3-9).
 This statement is, of course, consistent with our Lord’s teaching concerning Himself: namely that He is Lord of heaven and earth. Thus the Sabbath is his jurisdiction as ‘Lord’ and determined by Him alone. The Pharisees accused Christ of breaking the Sabbath, but only He knew how to truly live and practice it (cf. Matthew 12:1-12; Luke 4:16,31; 13:10-16; John 5:9-18)
 Note how often the New Testament refers to this as being an act of recreation or something new: Romans 6:4; 2 Corinthians 5:17; Galatians 6:15; Ephesians 4:22-25; Colossians 3:9-10; 1 Peter 1:3; Titus 3:5. Since the Sabbath is reflective of creation (Genesis 2:2,3; Exodus 20:8-11) this new creation demands a new day.