Russia's foreign ministry accused Ukraine of deliberately provoking the incident, with spokesperson Maria Zakharova telling state television that Moscow will summon a senior Ukrainian diplomat to complain. Ukraine, along with almost every other country in the world, refuses to recognize that claim.
Three wounded Ukrainian servicemen have received medical assistance and their lives are not threatened, it said, adding that a criminal case has been launched over the Ukrainian violation of the Russian border. The boats were towed to a nearby port.
Moscow alleged that Ukrainian ships had been manoeuvring dangerously and ignoring its instructions with the aim of stirring up tensions.
Diplomats said the United Nations security council would meet on Monday at both countries' request to discuss the matter.
Each side blames the other, and the Ukraine government said it could declare martial law.
During a meeting of the NSDC, Mr Poroshenk called Russia's actions "unprovoked and crazy".
Trump has not commented on the incident.
Russian Federation blocked Kerch Strait with merchant marine tanker and thus, blocked the passage for three Ukrainian ships, including unarmed tug and two small river gun boats of GURZA-Class.
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"This is a very risky provocation, which requires particular attention and a special investigation", Russian presidential spokesman Dmitry Peskov told state media. "The Ministry of Foreign Affairs of Ukraine will take all appropriate measures of diplomatic and worldwide legal response", it says.
Ukraine has pledged to respond to the incident.
North Atlantic Treaty Organisation issued a similar statement, saying: "We call for restraint and de-escalation". "NATO fully supports Ukraine's sovereignty and its territorial integrity, including its navigational rights in its territorial waters".
"Nato. should send in naval ships in the Sea of Azov", the Atlantic Council's Anders Aslund also said.
The US reacted by calling an emergency session of the UN Security Council in NY on Monday morning. It reopened the route early Monday.
Relations between the countries have gone steadily downhill since Russia's 2014 annexation of the Crimea peninsula. Moscow's military intervention in the peninsula and subsequent referendum inflamed separatist sentiment in eastern Ukraine, effectively triggering a rebel insurgency by pro-Russia forces.
Shipping via Kerch Strait nevertheless, as of 1700 UTC, remains blocked, already creating huge queues of merchant ships both in Azov and in Black seas.
Videos coming from the scene show a cargo ship, accompanied by two Russian military boats blocking the passage under the arch of the Kerch Bridge, which connects Crimea to Russia.
Ukranians reacted to the maritime standoff with anger.