No...and yes.
Let's start at the beginning. Genesis 2:3 So God blessed the seventh day and made it holy, because on it God rested from all his work that he had done in creation. There is something amazing about the seventh day of creation. Perhaps this is the foundation of why the number seven is so significant in the Scriptures. It is fitting that this number is seen as a number of fullness and completion because that is exactly what God is resting in at the end of the creation week. In various places throughout Scripture we find explicit reference to Genesis 2 as the foundational understanding for the whole concept of shabbat - REST.
The ten commandments given in the law to Israel would require everyone to rest on the seventh day every week (Exodus 20:8-11). The reason that this would happen every seventh day (every sabbath) was because God had created in six days and rested on the seventh. Every seventh day was a testimony to the fact that this people were a people called to rest in God as he dwelt with them.
As you read through the first five books of the bible, you will find Israel called to rest their land every seven years. After working the land for six years, it needs rest. This is a time to trust in the Lord's provision. Leviticus 26:34 "Then the land shall enjoy its Sabbaths as long as it lies desolate, while you are in your enemies' land; then the land shall rest and enjoy its Sabbaths. The land that they were to possess was to be a picture of rest. Once the enemies of God's holiness were removed, the people of God in the land of God were to be a picture of resting in their God and King.
When Solomon was building the temple, it was called a place of rest for God's dwelling presence with his people. David was not able to build the temple because he was depicted as a man of war. When victories had been won and peace was obtained, God gave this responsibility to his son Solomon. 1 Chronicles 28:2 Then King David rose to his feet and said: "Hear me, my brothers and my people. I had it in my heart to build a house of rest for the ark of the covenant of the LORD and for the footstool of our God, and I made preparations for building." It took Solomon seven years to build the temple (1 Kings 6:38), and when Solomon was dedicating the temple he made seven petitions to God (1 Kings 8).
As you read through the history of Israel we gain insight into the significance of the temple. The temple was the dwelling place of God. God rested in the temple. The temple was where the worship of the people would be directed as a kingdom under God's ultimate Kingship (they were a theocracy). The temple is the central importance of the kingdom, and Solomon's kingdom would be established forever IF he continued in strength by keeping God's commands and rules (1 Chronicles 28:4-7). Because Solomon fails in this, we are always looking for one who will truly obey God and be the ultimate fulfillment of all that the temple represents. We continue to look for a kingdom of rest that truly rests in God as he dwells with his people.
While Israel were to continue to keep the sabbath every seventh day, they would be a continual testimony of the fact that God has created us to rest in him. They would also be a continual testimony of the fact that every week this looked forward to One who would ultimately bring forever rest. The penalty for not keeping the sabbath was death (Exodus 31:14). Everyone who lived as an unbeliever, unwilling to reflect God's promise of rest, would die.
If you have been paying attention you might see how all of this finds fulfillment in Jesus. Jesus came and in his finished work fulfilled all that the temple represented. He IS the dwelling presence of God. He IS the mediating sacrifice for all humanity as the Great High Priest. He did finish this work on the cross and then was raised and seated in rule and reign at the right hand of God (Ephesians 1:20-23). Jesus has finished the work of bringing the fulfillment of God's rest in dying and rising to reconcile creation. And when was Jesus raised to this new life securing a new creation for all in him? Not the seventh day of the week, but the first. The fulfillment has been accomplished and now the new has come.
There is no doubt that rest conveys the finished work of God who takes his place to dwell with humanity in rule and reign over his creation. This is exactly what the law pointed to in Christ and exactly what Christ has completed in fulfilling that law.
Is Sunday the new sabbath rest? Well, from a perspective of law and keeping days, no. It has been fulfilled. From the perspective of Jesus having finished his work to rule and reign in his kingdom - yes. We gather on the first day of the week, resurrection day, because it is a day of fulfillment, not because it is a day of replacement of legal requirements. We testify as we meet as the church that we have rest in Christ who IS our rest, and we point to the consummated rest that we are yet to enjoy. I am not a Sabbatarian in the the normal use of the sense of the word and I think it is an unhelpful term. I am, however, a big believer that every Lord's day, on a Sunday, the church displays our rest in the fulfillment of rest in Jesus Christ; our risen and resting King. We display that we rest in our Creator who created all things in six days and rested on the seventh in completion of his work- a work fully and finally completed in Christ.