James Tissot (†1902)
When Christ became man He established the New Covenant as was foretold by the prophets. "Behold the days shall come, saith the Lord, and I will make a new covenant with the house of Israel, and with the house of Juda" (Jeremiah 31:31). However, the New Covenant did not abolish the Old Covenant, it fulfilled it. In Matthew 5:17 Jesus says "Do not think that I am come to destroy the law, or the prophets. I am not come to destroy, but to fulfill." What is the fulfillment of the Law? Jesus explains this to the Pharisees in Matthew 22:34-40: "34 But the Pharisees hearing that he had silenced the Sadducees, came together: 35 And one of them, a doctor of the law, asking him, tempting him: 36 Master, which is the greatest commandment in the law? 37 Jesus said to him: Thou shalt love the Lord thy God with thy whole heart, and with thy whole soul, and with thy whole mind. 38 This is the greatest and the first commandment. 39 And the second is like to this: Thou shalt love thy neighbor as thyself. 40 On these two commandments dependeth the whole law and the prophets." The Catechism of the Catholic Church (2053) explains this in other words: "The Law has not been abolished, but rather man is invited to rediscover it in the person of his Master who is its perfect fulfillment".
What about the Ten Commandments? Recall the earlier statement that the judicial and ritual precepts of the law were only binding on those to whom they were delivered for that period in time. The reason the Ten Commandments are still binding for Christians is because they are moral precepts, not judicial or ritual. However, there is an important distinction to make here! They are not binding simply because they are a moral precept of the Old Law, they are only binding because they are part of the natural law which applies to all humans across time and space. The natural law is rooted in reason. We know by reason alone that murder, for example, is an immoral act. In the words of St. Thomas Aquinas: "The Old Law showed forth the precepts of the natural law, and added certain precepts of its own. Accordingly, as to those precepts of the natural law contained in the Old Law, all were bound to observe the Old Law; not because they belonged to the Old Law, but because they belonged to the natural law. But as to those precepts which were added by the Old Law, they were not binding on save the Jewish people alone."
With this in mind, the criticism Catholics receive for worshiping on a Sunday and not Saturday is unfounded. Our obligation to worship God is discerned by reason alone and is part of the natural law. Worship on the Sabbath (Saturday) was simply a law imposed on the Jews for their own nation. We worship on Sunday because it was the day that Jesus rose from the dead. As the Catechism (2176) states: "The celebration of Sunday observes the moral commandment inscribed by nature in the human heart to render to God an outward, visible, public, and regular worship as a sign of his universal beneficence to all. Sunday worship fulfills the moral command of the Old Covenant, taking up its rhythm and spirit in the weekly celebration of the Creator and Redeemer of his people." This is just one of the unfounded criticisms of the Catholic Church based on a misinterpretation of the Old Law and its bearing on Christians.
Catholics, and most Christians, do not pick and choose which parts of the Bible to believe. Christians are not bound by the Old Law except for the moral precepts of that Law that contain elements of the natural law.