If the Supreme Court overturns Roe v. Wade then abortion would become illegal in America.
not true. If the court were to reverse itself (which it shouldn't), the 50 state legislatures would be free to pass laws banning the procedure. In addition, the U.S. Congress could forbid abortions. However this is unlikely because of the Senate's filibuster rule. There would need to be at least 60 Senators willing to allow a vote on the measure. In addition, while the House of Representatives is nominally pro-life, once the issue was really at hand, I would suspect many voters would elect pro-choice members.
There would be states that would ban abortions. Most likely the laws would have to have an exception for the life of the mother. In practice, the more conservative states would ban abortions, while the Democratic leaning states would offer those services. Some women in "deep red states" would probably seek illegal abortions. According to the research, the number of abortions wouldn't decrease even if it was outlawed, though the procedure would become more dangerous due to lack of professionalism.