How do Reformed Churches justify applying the 4th commandment to the Lord's Day? It is understood why Sunday is the Christian celebration every week, but it is not clear how the classically reformed justify applying the fourth commandment concerning the Sabbath, which was given to Israel with its own stipulations, to the Lord's Day. It would seem on its face that the Sabbath observance would be simply spiritualized as entering into the eternal rest, rather than moved to Sunday and being accompanied with the traditional worship service.
This a tricky yet very interesting question and I would vote +2 if I could.

Basically from what I understand, although not perceived as a strict law, in the same way that the Sabbath was instituted under the law of Moses, the argument was this:


A day of rest was made by God for man before Adam sinned (before the Mosaic Law) by calling the 7th day holy in creation.
That day was a pledge of the eternal rest promised to man, if he were to obtain righteousness and also a pattern to follow for his own benefit to rest and enjoy his own labours once a week.
The Mosaic institution and law further stressed the need to rest, or enter into God's rest, but now as a law of condemnation only spoke to the lack of rest and the real need to enter into a new rest by the Messiah, where God would rest in a new creation and men could enter into a real eternal rest by the works of Christ.
When the law ended by the death of Christ, a new day of rest had to be instituted because a day of rest was still needed.
To use the same day of rest after Christ would be misleading on two accounts; First, as the Law of Moses (ceremonial, moral/punitive, judicial - entire law) was removed in the death of Christ, the day must change to make it clear just as the curtains of the temple were torn to make it clear. Second, as Christ 'rested from his works' on the first day (Sunday) by his resurrection, having finished making a new creation, it was logically the day to chose by the Apostles.
Technically the day was not replacing the Sabbath as strict 'command under the Law' but replacing the original day of rest under the order of nature before the law, so was in harmony to grace. This may explain why although followed naturally we find no strict rules in the New Testament reading it.
In some sense although society needs a 'day' to organise itself under this rest, the new day is not necessarily so strictly 'legal' as we must remember it's purpose is for the benefit of man and a literal pledge of his eternal rest in Christ.


The best and most elaborate lengthly and reasonable explanation that I have seen made from a reformist view is by John Owen in "EXERCITATIONS CONCERNING THE NAME, ORIGINAL, NATURE, USE, AND CONTINUANCE OF A DAY OF SACRED REST: WHEREIN THE ORIGINAL OF THE SABBATH FROM THE FOUNDATION OF THE WORLD, THE MORALITY OF THE FOURTH COMMANDMENT, WITH THE CHANGE
OF THE SEVENTH DAY, ARE INQUIRED INTO; TOGETHER WITH AN ASSERTION OF THE DIVINE INSTITUTION OF THE LORD’S DAY, AND PRACTICAL DIRECTIONS FOR ITS DUE OBSERVATION BY JOHN OWEN:" (haha those old long titles hey?)

This quote is where Sunday is pinpointed as the day to fulfil the moral natural intent of the 4th command according to a traditional reformed view:


  
  First, on this day he rested from his works, in and by his resurrection; for then had he laid the foundation of the new heavens and new earth, and finished the works of the new creation, “when the morning stars sang together, and all the sons of God shouted for joy.” On this day he rested from his works, and was refreshed, as God did and was from his. For although he “worketh hitherto,” in the communication of his Spirit and graces, as the Father continued to do in his works of providence, after the finishing of the works of the old creation, though these works belonged thereunto, yet he ceased absolutely from that kind of work whereby he laid the foundation of the new creation. Henceforth he dieth no more. And on this day was he refreshed in the view of his work; for he saw that it was exceeding good. Now, as God’s rest, and his being refreshed in his work, on the seventh day of old, was a sufficient indication of the precise day of rest which he would have observed under the administration of that original law and covenant, so the rest of our Lord Jesus Christ, and his being refreshed in and from his works, on the first day, is a sufficient indication of the precise day of rest to be observed under the dispensation of the new covenant, now confirmed and established.
  And the church of Christ could not pass one week under the new testament, or in a gospel state of worship, without this indication; for the Judaical Sabbath, as sure as it was so, and as sure as it was annexed unto the Mosaical administration of the covenant, was so far abolished as not really to oblige the disciples of Christ in conscience unto the observation of it, whatever any of them might for a season apprehend. And if a new day was not now determined, there was no day or season appointed for the observance of a holy rest unto the Lord, nor any pledge given us of our entering into the rest of Christ. And those who say that it is required that some time be set apart unto the ends of a sabbatical rest, but that there is no divine indication of that time, when nor what it is or shall be, if we consider what are the ends of such a rest, as before declared, must allow us to expect firmer proofs of their uncouth assertion than any as yet we have met withal.
  Accordingly, this indication of the gospel day of rest and worship was embraced by the apostles, who were to be as the chief cornerstones, the foundation of the Christian church; for immediately hereon they assembled themselves on that day, and were confirmed in their obedience by the grace of our Lord, in meeting with them thereon, John 20:19, 26. And it seems that on this day only he appeared unto them when they were assembled together, although occasionally he showed himself to sundry of them at other seasons. Hence he left Thomas under his doubts a whole week before he gave him his gracious conviction, that he might do it in the assembly of his disciples on the first day of the week; from which time forward this day was never without its solemn assemblies, as shall further be cleared afterwards. (Owen, J. (1862). An Exposition of the Epistle to the Hebrews. (W. H. Goold, Ed.) (Vol. 19, pp. 409–410). Edinburgh: T&amp;T Clark.)