I feel bad that my joke post was the only response, so here's a real answer:
Even if he avoids those gaffes, I don't see Schweitzer running, because he always seemed to be fairly uncommitted about running for other offices (most notably: passing on running for the Senate), in large part because he liked retirement so much. Of course, President of the United States is a much more interesting and prestigious office, but then I think he'd go back-and-forth on it, agonize over it, before either declining or simply take too long to decide and lose key staffers in the process. The other problem, of course, is that 2016 is Hillary's Time, where every rising star in the party opted to sit it out in either deference or fear of her fundraising prowess— and I think Schweitzer probably seems that as an insurmountable wall, too.
If he does run, though, I don't see him getting much traction. Schweitzer got a fair amount of national attention during his governorship, but it didn't last after he was out of office, and by 2016 I think he's practically an obscure figure known only by pundits and politics-watchers. As far as becoming the "anybody but Clinton" choice, I think his political positions complicate that; because while he's got populist-progressive chops in opposing military intervention and supporting Medicare for All (before it was cool), he also is pro-gun (he had an A rating from the NRA!) and pro-pipeline, and was notably guarded and taciturn on his view on same-sex marriage— which could easily derail his popularity with the progressive grassroots that elevated Sanders. And that's not even getting into the possibility of him making gaffes during the primary season… if he said things along the lines of those "gaydar" or Feinstein comments, his insurgency is done.
In short: Schweitzer's views make him an interesting and potentially formidable candidate, but 2016— at least as it stands IOTL— makes him a non-starter, I think. In a world where 2016 is an "open" nomination (no Clinton, no Biden), I can see him doing well.