Ukraine is not an official member of North Atlantic Treaty Organisation, but the alliance has given support to Ukraine in the past.
The president enacted martial law for some provinces for a limited period after parliamentary support.
"Putin wants the old Russian empire back", he said.
President Poroshenko said the global community must not accept Russia's aggressive policy, in an interview with German newspaper Bild.
The religious rift follows the broader collapse in relations between the two neighbours following Russia's seizure of Crimea in 2014 and backing of separatist uprisings in eastern Ukraine that have killed more than 10,000 people.
"We condemn this demonstration of barbarism and are multiplying our efforts to bring our boys home", Ukrainian Prime Minister Volodymyr Groysman said on Twitter after the court rulings. Ukraine insisted its vessels were operating in line with worldwide maritime rules in the strait, which separates Russia's mainland and the Crimean Peninsula that it annexed from Ukraine in 2014.
U.S. Defense Secretary Jim Mattis said Wednesday that Russia's seizure of the Ukrainian ships violated a treaty between the two countries and showed that Moscow can not be trusted.
In recent months, Russian Federation has been deploying more forces in and around the Sea of Azov, stopping and searching almost 300 commercial vessels in an attempt to pressure Ukraine economically. Trump, who was en route to Buenos Aires for the G-20 summit, said he would not be meeting Putin because "the ships and sailors have not been returned to Ukraine from Russian Federation". This though specifically states that warships from third countries can only enter the Sea or make port visits there with the express permission of the other party.More news: US, China Exploring Deal to Delay More Tariffs, Start Talks
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The announcement likely ratchets up tensions further and could cause significant disruptions for Russians traveling for work or to see family. That followed a warning from Poroshenko that Ukrainian intelligence services had ascertained Russian Federation was amassing tanks and ships across the border and there was a "threat of full-scale war".
The United States and other countries have also criticized the Russian actions in the Kerch Strait.
Russia's decision to open a bridge across the Kerch Strait earlier this year had already exacerbated tensions.
In his first extensive remarks since the confrontation at sea on Sunday, Putin said it had been orchestrated by Kiev as a "provocation".
Millions of Russians have family members in Ukraine and vice versa as both countries were formerly part of the Soviet Union. Prior to the annexation of Crimea, his approval rating had dropped to record lows; afterwards, it surged to more than 80%. Recent polls suggest that only about 10% of the electorate plans to vote for him next year, with almost 50% saying they would not vote for him under any circumstances, the Kyiv Post newspaper reported.
Putin insisted that Russia's military response was appropriate as the Ukrainians had "trespassed" into Russia's territorial waters.
Meanwhile, a Russian government-appointed ombudswoman in Crimea said the captured Ukrainian naval personnel are being transferred to Moscow, Russian state media reported on November 30.
But the incident has prompted renewed calls for more Western sanctions on Russian Federation. According to Ukrainian officials, the ships came under fire and one boat was rammed by a Russian ship.