Russia said it was a violation of its borders; Ukraine described it as a routine passage.
Russian news agencies reporting from a Crimean court on Tuesday quoted an investigator as saying that the man faces charges of violating the border, which carry up to six years in prison.
The Ukrainian was ordered held until January 26.
It comes as the Kremlin warned that Ukraine's declaration of martial law over the incident might trigger a flare-up in hostilities in eastern Ukraine, while Kiev blamed Russia for parading captured Ukrainian seamen on television.
Russia and Ukraine have blamed each other for Sunday's confrontation in the Kerch Strait, which links the Black Sea and the Sea of Azov.
The clash has raised the spectre of renewing a full-blown conflict in eastern Ukraine and saw Russia strongly criticised at the United Nations by the United States and its allies.
The Ukrainian parliament on Monday adopted a motion by the president to impose martial law for 30 days.
Defence Secretary Gavin Williamson recently announced he was to commit more troops to Ukraine as well as send HMS Echo, a hydrographic survey ship, to the Black Sea.
The Ministry of Defence said the vessel's presence aimed to "demonstrate the UK's support to ensuring freedom of navigation".
After the latest seaborne confrontation, Lord West said: "The minister will be aware that it has been stated that we are sending a warship to Black Sea.
"That warship as I understand it is actually a survey ship.
"If things are hotting up in the Black Sea to send a ship in harm's way that is not really capable of looking after itself is not a clever idea.
"Should this reviewed and perhaps we should send a ship like the [Type] 45 that is able to look after itself in these circumstances."
Tory frontbencher Baroness Goldie said: "I am very reluctant to comment on specific operational matters.
"The MOD response to such situations is carefully assessed, carefully reviewed, and any decision to deploy our ships would only be made after the most careful assessment of all the circumstances."
Liberal Democrat peer Lord Wallace of Saltaire said: "If the Russians succeed in converting the Sea of Azov into an internal sea that would have a devastating effect on the economy of a substantial chunk of Ukraine because of the port of Mariupol."
Lady Goldie said: "Regrettably I think these developments represent a step change in the creeping annexation by Russia of the Sea of Azov.
"Indeed Russia's use of force, including use firearms against Ukraine's vessels, marks clear aggression.
"The actions are a breach of international law."
She added: "The United Kingdom and our allies have been very clear in our profound disquiet at that action."
Former UK ambassador to the UN Lord Hannay of Chiswick said: "This act by the Russians is not like all the other things they do in eastern Ukraine - deniable in any way - because it is an act by the Russian navy.
"Is it not also the case that to use force to enforce a blockade is, in fact, an act of war."
Lady Goldie said: "It's certainly an act of aggression and it is a further example of Russia's ongoing violation of Ukraine sovereignty and territorial integrity."