Ukrainian officials say at least six sailors were wounded and denies doing anything wrong, accusing Russia of military aggression.
"Such actions pose a threat to the security of all states in the Black Sea region and therefore require a clear response from the international community," Ukraine's foreign ministry said.
The Russian Federal Security Service (FSB) accused Ukraine of staging a deliberate provocation.
Ukrainian President Petro Poroshenko has signed a bill Monday to impose martial law in the country, in the wake of Russia seizing three of its ships and their crews.
The bill still must be approved by parliament.
Russia fired on two Ukrainian naval ships and rammed a third vessel Sunday in the Black Sea, seizing the ships and accusing them of illegally entering its territorial waters.
Poroshenko said Monday he wants to declare martial law "to strengthen Ukraine's defense capabilities amid increasing aggression and according to international law a cold act of aggression by the Russian Federation."He added that Ukraine intends "to keep adhering to all international obligations."
Poroshenko is demanding Russia immediately release the Ukrainian sailors and ships.
Russian Foreign Minister Sergei Lavrov accused Ukraine of violating international norms with "dangerous methods that created threats and risks for the normal movement of ships in the area." Russia continued to ignore Western calls to release the sailors Sunday.
The U.N. Security Council is holding an emergency meeting to address the situation in an effort to keep it from escalating.
European Union chief Donald Tusk condemned Russia's use of force and reiterated the EU would stand in support of Ukraine.
"Russian authorities must return Ukrainian sailors, vessels & refrain from further provocations," he said in a tweet Monday, adding he had spoken with Poroshenko.
The Kerch Strait is the only passage between the two seas.
Ukraine said Russia used a tanker to block access to the Kerch Strait, which under a treaty is shared territory.
Russia said the Ukrainian ships were violating its waters and accuses the Ukrainians of failing to inform it that three of its ships were planning to sail through Kerch — a charge Ukraine denies.
Both NATO and the European Union issued statements late Sunday urging both sides to act with the "utmost restraint" while NATO calls on Russia to "ensure unhindered access to Ukrainian ports in the Azov Sea in accordance with international law."
The Trump administration has previously warned Russia against trying to strangle the Ukrainian economy by harassing international shipping through the Kerch Strait.
Russia forcibly annexed Ukraine's Crimea Peninsula in 2014, claiming its ethnic Russian majority was under threat from the Ukrainian government.
Fighting between pro-Russian forces in eastern Ukraine and Ukrainian troops has eased in recent months, but there are still occasional deadly flare-ups.
Russia has consistently denied sending weapons and fighters to help the separatists, despite strong evidence to the contrary.
