If Obama really wants to be serious (and do more than talking about) responding to today's SCOTUS decision, he will propose a plan to expand the number of seats on the Supreme Court. No, the ones already there won't like it, but that's one way to give Obama at least two immediate appointments to that body. That would be an expansion to 11. But, one could easily go as high as 15. I say start at 17, and negotiate down to 13, but that decision would really be up to Congress.
We can't remove the ones who are there, but we can do something that needs doing anyway: expanding the supreme court to better reflect the breadth and diversity of our society. It also means that no single president will be able to completely shift the Court to the fringes, even if they serve two terms and get several appointments. I think that's a prudent idea, given the current Court and the ideological disaster that was the previous administration.
Yes, they will scream "court packing" like they did with Roosevelt. But, I'll tell you this; There will never be a better opportunity for making fundamental changes at the Court. Responding on this issue crosses party lines, and such a plan can now be sold to the public as a response.
Unlike undoing the actual precedent set by the court today (which WILL require a constitutional amendment since it was grounded in constitutional considerations, thus requiring a supermajority in BOTH houses of Congress to pass), changing the seats on the court can be done by simply changing the law. That means a simply majority.
Of course, there is that nasty filibuster thing in the Senate that we have to worry even more about now. But, I would relish watching the Republicans (and, let's face it, Joe Lieberman) try to defend the "rights" of corporations to take over our political system outright. That will be some Must See C-Span, which doesn't happen often.
Anyway, if you want a standard to measure the Democrats actual response on this new decision, there it is. Those are their actual options for action: Constitutional Amendment and change the seats on the Court.