Why do Protestants worship on Sunday?

The Roman Catholic Church did change the Sabbath from Seventh Day to the First Day.

This is taken from here:

Q. Why do we observe Sunday instead of Saturday?
A. We observe Sunday instead of Saturday because the Catholic Church transferred the solemnity from Saturday to Sunday."

-- The Convert's Catechism of Catholic Doctrine, 1951 printing, page 50.

Q. How prove you that the Church hath power to command feasts and holy days?
A. By the very act of changing the Sabbath into Sunday which Protestants allow of; and therefore they fondly contradict themselves, by keeping Sunday strictly, and breaking most other feasts commanded by the same church."

-- Henry Tubervill, An Abridgment of the Christian Doctrine, 1833, page 58.

There are many protestant groups who keep the True Seventh-Day Sabbath as the seventh day (saturday). The chief among them are Seventh-Day Adventists and Seventh-Day Baptists.

The reason for most protestants not keeping 7th day sabbath is that the truth about Sabbath was re-discovered much later than when the originators of the respective churches lived.

And the times of this ignorance God winked at; but now commandeth all men every where to repent: Acts 17:30

Robinson’s Farewell Address to the Pilgrims upon their Departure from Holland, 1620. The Account by Edward Winslow in his “Hypocrisie Unmasked,” Printed in 1646:

And if God should reveal anything to us by any other instrument of his, to be as ready to receive it, as ever we were to receive any truth by his Ministry: For he was very confident the Lord had more truth and light yet to breake forth out of his holy Word. He took occasion also miserably to bewaile the state and condition of the Reformed churches, who were come to a period in Religion, and would goe no further then the instruments of their Reformation: As for example, the Lutherans they could not be drawne to goe beyond what Luther saw, for whatever part of God’s will he had further imparted and revealed to Calvin, they will rather die then embrace it. And so also, saith he, you see the Calvinists, they stick where he left them: A misery much to bee lamented; For though they were precious shining lights in their times, yet God had not revealed his whole will to them: And were they now living, saith hee, they would bee as ready and willing to embrace further [2] light, as that they had received. Here also he put us in mind of our Church-Covenant (at least that part of it) whereby wee promise and covenant with God and one with another, to receive whatsoever light or truth shall be made known to us from his written Word: but withall exhorted us to take heed what we received for truth, and well to examine and compare, and weigh it with other Scriptures of truth, before we received it; For, saith he, It is not possible the Christian world should come so lately out of such thick Antichristian darknesse, and that full perfection of knowledge should breake forth at once.

For the full speech see here.

To put it simply, the Sabbath (as per Mosaic law is Friday Sundown to Saturday Sundown). It is written in Acts that readings were given on the first day of the week (Sunday) and the Catholic Church adopted this as the Sabbath. (Readings were probably given every other day as well!) The Sabbath was made for man to rest and for worship/prayer but we are not under the Law but grace, so what is in the heart is all important. I personally try to Observe The Old Sabbaths but like I said whatever ordinances we observe, as long as it is of faith is the main.

The choice of Sunday is traditional, but it is also harmonious with the Biblical account (emphasis added):

Acts 20:7 NASB
On the first day of the week, when we were gathered together to break bread, Paul began talking to them, intending to leave the next day, and he prolonged his message until midnight.

1 Corinthians 16:2 (NASB)
On the first day of every week each one of you is to put aside and save, as he may prosper, so that no collections be made when I come.

Revelation 1:10 (NASB)
I was in the Spirit on the Lord's day, and ...

There's a strong suggestion¹ here that Sunday was day that the church met together. Because it was the day of Jesus' resurrection (Mt 28:1), it seems very sensible for the modern church to do this.

Also, bear in mind that there is no commandment from Jesus or the Biblical authors to worship or meet together on a particular day of the week. If one was not bound by the Law of Moses, why would it matter what day was chosen? Not all protestant groups believe that they are not bound, but of those who do, what compulsion would there be to worship on Saturday except for tradition and honor for the Jewish way?

¹ In regards to the relevance and validity of including a passage about "the Lord's Day," I include the passage because is a possible example of Sunday having its own significance. It seems reasonable (though not the only conceivable reading) to understand this as Sunday because John used the word Sabbath nine times, never once referring to it as the Lord's day or giving it any other name. (The particular phrase doesn't appear anywhere else in the NT.) No other NT author referred to the Sabbath in any other way than the name already employed by the Jews. Revelation was presumably written later than his other writings, and so perhaps his understanding was better, or the phrase grew out of what was by that time established tradition (60-70 years after Jesus' death) of whatever day the Christians met together. To choose between the two most likely meanings, it seems to me that Sunday has a stronger case than Saturday.