An interesting question considering that if you count daily Mass and the Liturgy of the Hours, Catholics worship on every Saturday and Wednesday, as well as every other day of the week. By Saturday worship I assume you’re talking about the Seventh Day Adventists who celebrate the Sabbath on Saturday. This practice comes from their acknowledgement that it on the seventh day of the week that God rested. Thus, the historical developments of the early Church notwithstanding, they claim that they have simply recovered what should be the true Sabbath and so they act accordingly.
The practice of many protestant communities having devotional services on Wednesday grew out of the practice of setting aside Wednesday as a day for mid-week instruction in the faith. While many of these instructional sessions have elements of prayer and praise, they are not the same as Sunday Sabbath worship. It can get confusing though since for many Evangelical Christians, the term “worship” is used to denote any time that Christians come together in prayer and praise of God.