The Supreme Court is independent, yes, but Congress has several options for exercising control over the justices.
Wilson Huhn has posted an article discussing in detail the likely effects if President Obama were to have the chance to replace any of the five nonliberals on the Supreme Court. Huhn argues that a large number of cases would be likely to be overturned. He basically divides up the cases into those that he believes will be overruled and those who believes will probably or might be overruled. I generally agree with his assessments.
In the category of those he predicts will be overruled are:
Citizens United v. FEC (protecting under the First Amendment contributions from corporations and unions);
Zelman v. Simmons-Harris (allowing government to provide vouchers for children to attend parochial schools);
Van Orden v. Perry (allowing Texas to place the Ten Commandments on the state capitol under circumstances suggesting the non-religious message predominates);
NFIB v. Sebelius (five members of the Court concluding that the Commerce Clause does not allow Congress to mandate the purchase of health insurance)
There are a greater number of cases in the category of might or probably would be overruled, including
The cases allowing political gerrymandering;
Heller and McDonald (applying an individual right to keep and bear arms outside of the militia context);
Lorillard v. Reilly (holding Massachusetts ban on certain tobacco advertising was unconstitutional).
Claims of this type about the effects of a presidential election on constitutional law are often made. But they often do not pan out either because the nonliberal justice does not step down or other circumstances intervene (for example, even if the nonliberal justice is replaced by a liberal a Republican President might be elected in 2016 and a liberal justice might then be replaced).
Still, what is striking about the article is how many cases are predicted to be overruled. The Supreme Court claims to follow precedent, but their precedent rules do not do much to constrain them, especially as to controversial cases.