The other day, I ran through the various options and concerns and issues regarding potential military action in Syria. In doing so, I noted that the United Nations route was basically out of play, as Russia had stopped it up. Going in alone, though, just the United States acting on the matter, is a bad idea, even if the action is limited. Historically, that tends to not turn out well; matters are often made worse. Any action we take would need allies, even if only for the purposes of optics. France wishes to go, but that's not enough.
By far, the best option- the only option, really- if you were looking for a sufficiently-sized coalition without the UN's involvement is to go to NATO, as happened in Syria. But doing so requires unanimous support from all 28 NATO countries, and after the United Kingdom, a NATO member, declined to involve themselves, that was it for that option. NATO has officially opted out.
Unfortunately, I think that's settled the matter. The decision has essentially been made for us. It can't just be us and France. The question is not should we go in. Something SHOULD be done. This SHOULDN'T, morally, be allowed to continue. That's not the problem. The problem, the question, is CAN something be done. Can this situation be made better with our involvement, or is this somehow still the best-case scenario in comparison to what would happen if we acted. Can this be done without Iran firing at Israel and Russia firing at Saudi Arabia and everyone involved in the Middle East shooting at everyone else in a redux of World War 1.
The answer, much as I hate to say it, appears in the absence of NATO to be: no. No, it can't.