DONETSK, Ukraine—Pro-Moscow protesters occupying government buildings in eastern Ukraine called Monday for their own referendum on independence, a flare-up that spurred a renewed diplomatic push and a fresh warning by the U.S. against any Russian escalation.

U.S. Secretary of State John Kerry, after a phone call with Russian Foreign Minister Sergei Lavrov, announced he would meet together with Russian, Ukrainian and European officials within 10 days. In an apparent glimmer of progress, Mr. Lavrov and his Ukrainian counterpart,...