On Sunday Russian border guards fired on three Ukrainian ships and seized their crews off the Crimean Peninsula.
Mr Trump said he would not meet Mr Putin at a G20 summit in the coming days, “based on the fact that the ships and sailors have not been returned”.
German Chancellor Angela Merkel blamed the crisis “entirely” on Russia.
She said she would raise the issue with President Putin at the G20 meeting, which is due to be held in Argentina on Friday and Saturday.
Ukrainian President Petro Poroshenko has urged Nato to send ships to the area. He has implemented martial law across Ukraine’s border regions for 30 days in response to the crisis.
On Thursday he announced that Russians living in Ukraine would soon face restrictions on bank withdrawals, changing foreign currency and travelling abroad.
The incident happened on Sunday, when two Ukrainian gunboats and a tug were sailing from Odessa to the port of Mariupol, in the Sea of Azov – which is shared between Russia and Ukraine.
The vessels were stopped from entering the Kerch Strait and confronted by FSB border guards. After a lengthy standoff, during which the Ukrainian tug was rammed, the ships began turning back towards Odessa, the Ukrainian government says.
The Russians opened fire, wounding at least three sailors, and seized the Ukrainian flotilla.
The Kerch Strait separates Russia from Crimea, a Ukrainian peninsula that was annexed by Russia in 2014.
Ukraine says Russia is deliberately blockading Mariupol and another Ukrainian port on the Sea of Azov, Berdyansk.
The 24 captured Ukrainian sailors have now been given two months in pre-trial detention by a court in Crimea.