That's war and things happen. I did not read the exact statement of the US representatives, but even in the Russian media it was said that they expressed regrets, if not apologies. IMHO, that's more than enough in this particular situation, and the others in the US-led coalition may not elaborate more as they were obviouly getting the target information from US.
Syrian government can file the protest, and likely will, otherwise it would lose the "remainders of its face". But it does not control a significant part of its territory and better shut up letting the others wipe the garbage out of their territory.
UN is toothless, and the only thing that really matters is to make real forces operating in this filed (read, US/NATO and Russia) talk to each other and coordinate their activities. Russia is likely having better ISR as they are operating in Syria (i.e. not coming from Iraq like the coalition does) and Syrian land forces (who are reporting to them) can obviously better locate small Daeash groups than UAVs and sats. The coalition has more hi-pres weapons on the theatre, so there might be a place for "synergy".
There were some signs of hope that such a coordination plan was being worked out not long ago, but it was not disclosed and thus it's difficult to judge. I wonder why it is kept in confidence. Were there anything about kurds that would make Erdogan angry (which neither Russia, nor US are interested in)?