Yeah the history of this is long and complicated. This piece goes into a lot more detail about the various attempts on both sides to put abortion into federal legislation (both for and against), including a review of the political situations that led to and resulted from each effort. One of the more interesting threads is this one:

After [Joe] Biden joined the Senate in 1973, he voted for a failed constitutional amendment that would have allowed states to overturn the court’s Roe ruling. In a Washingtonian magazine interview at the time, he said of Roe: “I think it went too far. I don’t think that a woman has the sole right to say what should happen to her body.” But times changed, and so did he. In a 2007 book, Biden said he had arrived at a “middle-of-the-road position on abortion.” In 2008, he described Roe as “close to a consensus that can exist in a society as heterogeneous as ours.” As Obama’s vice president, Biden said the government had no “right to tell other people that women, they can’t control their own body.”

The cultural center of gravity on this issue has changed dramatically since the 1970s. There were pro choice voices in both parties for many decades, but with very few exceptions those voices have migrated into the Democratic Party, which gives it a more unified and consistent stance on the issue.