I think there are many reasons to resist such calls, but in this post I want to focus on one: I think such suppression would likely lead to more riots and more deaths, not less. Here’s why.
Behavior that gets rewarded, gets repeated. (Relatedly, “once you have paid him the Dane-geld, you never get rid of the Dane.”) Say that the murders in Libya lead us to pass a law banning some kinds of speech that Muslims find offensive or blasphemous, or reinterpreting our First Amendment rules to make it possible to punish such speech under some existing law.
What then will extremist Muslims see? They killed several Americans (maybe itself a plus from their view). In exchange, they’ve gotten America to submit to their will. And on top of that, they’ve gotten back at blasphemers, and deter future blasphemy. A triple victory.
I've been thinking a little more about the riots. There is an instance here of a more general problem: often the right course of action requires more sacrifice or even makes things worse on some measures in the short term.
For example, saving money makes you less well off now in consumption terms, but better off in the long run. When FDR and Congress declared war on imperial Japan in the wake of Pearl Harbor, it made things much worse on a short-term perspective, as any Marine on the beach at Iwo Jima could testify.
People are often very bad at making trade-offs in time or discounting the future. Still, the right answer to this problem is not necessarily the one that calms frenzied Islamic mobs in the short term.
Yes, responsible leaders ought to try to damp down short-term emotions. But then they also need to think rationally about the situation, rather than engage in wishful thinking.
Much of the problem in the Middle East is inherent conflicts are avoided or kept simmering without resolution. Volokh cites Pew research which finds more than three-quarters of Egyptian Muslims support the death penalty for apostasy, I.e leaving the Muslim religion. The Arab Street is deeply hostile to and profoundly at odds with our way of life. It is possible to try to damp things down too much.
There is little upside to compromsing our own principles to try to buy favor with Islamic mobs. If it is not a stupid video today, it will be rioting for the return of Al-Andalus to Islam tomorrow.