Troops are surrounding Ukrainian military bases and other installations, while two Ukrainian warships are reported to be blocked by a Russian ship in the port of Sevastopol.
Ukrainian TV reported on Tuesday evening that armed men had attempted to take over an anti-aircraft missile base in Yevpatoria, on the coast north of Sevastopol.
Kiev and the West have alleged Russia is mounting an invasion of Crimea, which has a majority Russian-speaking population.
Mr Obama accused Russia of "seeking through force to exert influence on a neighbouring country".
"That is not how international law is supposed to operate," he said.
But at a lengthy news conference on Tuesday, in his first public comments on the issue, Mr Putin denied the heavily armed troops were Russian. He said they were "local self-defence forces" loyal to Moscow, protecting the bases from "nationalists" and "anti-Semites".
Russia, said Mr Putin, reserved the right to act to protect Russian citizens and speakers anywhere in Ukraine.
In Crimea and in eastern Ukraine, there have been shows of support for Russian intervention.