I think it was more a call for stability than anything else. Given the long border Turkey has with Syria and Iraq, I assume their number one concern was that anything that destabilized Turkey could have incredibly serious consequences -- with ISIS moving into a country that is half in Europe.
Kerry had been in intensive talks with Lavrov all day, following long talks that they had the previous day also including Putin. His comment at the news conference in Moscow, first said that both he and Lavrov were just learning about Turkey.
Both Obama and Kerry had the addition concern - if that was not enough - that there are many American military people at the base that they have used to fight against ISIS - which now has the power out and they are not allowed to fly. I assume that this has constrained their comments.
Business Insider tends to RW/neo con oppinions. Here, their editorializing "clear support" rather overstates what is a pretty minimal statement -
"The President and Secretary agreed that all parties in Turkey should support the democratically elected government of Turkey, show restraint, and avoid any violence or bloodshed. The Secretary underscored that the State Department will continue to focus on the safety and security of US citizens in Turkey. The President asked the Secretary to continue to keep him updated as the situation unfolds."
Kerry also issued a separate statement of support, adding that the US was viewing the situation with "gravest concern."
Note that restrain and avoiding violence is not applied to just one side. You could say that firing scores of judges is NOT restraint.