It said the Mu Du Bong was a legitimate commercial ship and its detention a “rampant violation” of sovereignty.
But a UN expert says the ship belongs to North Korea’s Ocean Maritime Management, which is on a UN blacklist.
In July 2013 one of its ships was seized in Panama after Soviet-era weapons and fighter jets were found hidden under sugar sacks.
United Nations sanctions ban most arms shipments to North Korea.
Under resolutions adopted after Pyongyang’s nuclear tests in 2006 and 2009, the export of all arms and related parts, with the exception of small arms and light weapons, to the communist country is prohibited.
The Mu Du Bong ran aground on a reef off Mexico’s Veracruz state in July 2014. North Korea says it has since paid a bond to cover damage to the reef.
Speaking at a news conference at the UN on Wednesday, North Korea’s deputy ambassador An Myong Hun said the ship and its crew should be released.
“This ship is totally a peaceful and legitimate commercial ship which sails under the direction of the Ministry of Land and Sea Transportation,” he said.
“The detention of Mu Du Bong is a rampant violation of the dignified sovereignty of the DPRK [North Korea].”
But Hugh Griffiths, co-ordinator of the UN panel that oversees sanctions violations, told journalists that there was “overwhelming” evidence to show the ship was linked to OMM.
A spokesman for Mexico’s UN mission, meanwhile, told AFP news agency his country was “fulfilling our international obligation under Security Council resolutions”.
OMM was blacklisted by the UN in July 2014. The UN said it had played a key role in arranging the shipment of concealed arms found in Panama.
The Chong Chon Gang was stopped on the Atlantic side of the Panama Canal on 15 July 2013 over suspicions it was carrying drugs.
It had disappeared from satellite tracking for a few days as it approached the Cuban capital, Havana, having departed from Russia’s eastern coast three months earlier.
On searching the vessel, officials found military hardware including two Soviet-era MiG-21 fighter aircraft, air defence systems, missiles and command and control vehicles.
Cuban authorities said that the ship was carrying 240 tonnes of “obsolete” defensive weapons.
The North Korean government insisted the ageing weapons were simply being transferred to North Korea to be repaired, before returning them.
But the US envoy to the UN, Samantha Power, described the episode as a “cynical, outrageous and illegal attempt” by Cuba and North Korea to circumvent UN sanctions.