I actually think Gomez was playing his very best for us at left back under Rodgers, before he suffered a nasty injury. It was for a short period, and no, I don't think he should be a left back in the long term, but I do remember thinking he was sensational at the time.
I do think he'll develop as a CB because he seems to be more naturally inclined to defend, and though he's comfortable on the ball, he doesn't really offer that much going forward.
The trend in recent years has been of seeing quite a few fullbacks converted into central defenders in a 3-man backline (Azpilicueta and Monreal, for instance), and on occasion we've seen wingers playing as fullbacks or wingbacks, and in that environment I wouldn't expect a more defensive player like Gomez to be a long term option as a right back. But who knows, I think he's talented enough to have a decent career at more than one position.