Politifact adjudicates the debate:

The Facebook post says Mitt Romney is wrong to claim that spending under Obama has "accelerated at a pace without precedent in recent history," because it's actually risen "slower than at any time in nearly 60 years."

Obama has indeed presided over the slowest growth in spending of any president using raw dollars, and it was the second-slowest if you adjust for inflation. The math simultaneously backs up Nutting’s calculations and demolishes Romney’s contention. The only significant shortcoming of the graphic is that it fails to note that some of the restraint in spending was fueled by demands from congressional Republicans. On balance, we rate the claim Mostly True.