Typical attempts to substantiate their claims involve the citation of various nineteenth century authors, who simply made guesses regarding the methodology of early Christian worship services. However, any notion of the Church of Rome changing the Sabbath Day to Sunday is false. Mandatory Sabbath observance was only meant for Israel. The tradition of gathering on Sundays for worship has been practiced since the first century in correspondence to the resurrection of Jesus Christ.
The text of Acts 20:7-12 very clearly describes Christians as having fellowship on Sunday. The New Testament records the existence of this tradition elsewhere in passages such as John 20:19-20, 1 Corinthians 16:1-2, and Revelation 1:10. If we are going to approach the New Testament as a historically reliable document, then that would mean Christians had already been worshiping on the first day of the week. Constantine made no such change to the Sabbath. The tradition of gathering on Sunday can be found in the earliest existing sources outside the New Testament:
"Christian Assembly on the Lord's Day: But every Lord's day do ye gather yourselves together, and break bread, and give thanksgiving after having confessed your transgressions, that your sacrifice may be pure. But let no one that is at variance with his fellow come together with you, until they be reconciled, that your sacrifice may not be profaned. For this is that which was spoken by the Lord: In every place and time offer to me a pure sacrifice; for I am a great King, saith the Lord, and my name is wonderful among the nations." (Didache: The Teaching of the Twelve Apostles, Chapter XIV)
"[T]he Gentiles, who have believed on Him, and have repented of the sins which they have committed, they shall receive the inheritance along with the patriarchs and the prophets, and the just men who have descended from Jacob, even although they neither keep the Sabbath, nor are circumcised, nor observe the feasts." (Justin Martyr, Dialogue with Trypho, chapter XXVI)
"[T]hose who were brought up in the ancient order of things [i.e. Jews] have come to the possession of a new hope, no longer observing the Sabbath, but living in the observance of the Lord’s day, on which also our life has sprung up again by him and by his death." (Ignatius of Antioch, Letter to the Magnesians 8 [A.D. 110])
It is simply invalid to assert that the Roman Emperor Constantine altered this Jewish day of observance from Saturday to Sunday. Nobody has the power to change the Sabbath because it was originally instituted by God for the Jewish people.