Obviously President Obama has the right to appoint a new Supreme Court nominee, but realistically this Senate is extremely unlikely to confirm any of his appointees. He has the right, but I'm not holding my breath that the process will be completed this term.
Hopefully, in my opinion, there will be a Democrat in the White House after 2016 election and maybe a friendlier mix in Congress. The process of things like filling Supreme Court positions has ground to a halt.
If the people, who show up to the polls in 2016, vote Republican, Rowe v Wade and gay marriage will likely be overturned. Hope liberal minded people show up, or vote absentee (as in Washington State) and vote for the Democrats. The ball gets bounced back and forth between Obama and the Senate, but really, the ball lands in the court of the American people. Remember to register and vote.