The Freedom Flotilla Coalition, an international non-governmental group, published video footage showing a fire on one of its ships, named the Conscience.
The NGO said: "Israeli ambassadors must be summoned and answer to violations of international law, including the ongoing blockade (of Gaza) and the bombing of our civilian vessel in international waters."
The Israeli foreign ministry did not immediately reply to a request for comment.
The ship issued an SOS distress call after being hit 32km east of Malta, to which Cyprus responded by dispatching a vessel, the charity said.
The Maltese government said maritime authorities had received a mayday call from a vessel reporting a fire shortly after midnight local time. The vessel was outside territorial waters and had 12 crew members and four civilians on board, the government said.
It said a nearby tug headed to the scene and launched firefighting operations and a Maltese patrol vessel was dispatched. After several hours, the vessel and its crew were secure, it said, adding that crew had refused to board the tug.
In an earlier social media post on the incident, the Freedom Flotilla Coalition had said the vessel was at risk of sinking with 30 international human rights activists on board.
The coalition is campaigning to end Israel's blockade of Gaza. It said it had been organising a non-violent action under a media blackout in order to avoid any potential sabotage.
"Volunteers from over 21 countries travelled to Malta to board the mission to Gaza, including prominent figures," it said in a fuller statement on its website.
"On the morning of their scheduled departure, the vessel was attacked. Armed drones attacked the front of an unarmed civilian vessel twice, causing a fire and a substantial breach in the hull. The last communication in the early morning of the 2nd of May, indicated the drones are still circling the ship."
It released video footage shot in the dark that showed lights in the sky in front of the ship and the sound of explosions. The footage also showed the vessel on fire.
The Gaza war started after Hamas-led fighters killed 1200 people and took 251 hostages to Gaza in the October 7, 2023 attacks, according to Israeli tallies. Since then, Israel's offensive on the enclave killed more than 52,000, according to Palestinian health officials.
Another coalition ship on a similar mission to Gaza in 2010 was stopped and boarded by Israeli troops, and nine activists died. Other ships have similarly been stopped and boarded, without loss of life.
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