The Sea-Eye 4 had already rescued 397 people in six operations by Wednesday, assisted by the Mission Lifeline ship Rise Above, Sea-Eye said in a statement.
The vessel then picked up a further 400 people from a heavily overcrowded wooden boat on Thursday morning, it said.
Maltese rescue services, though responsible for maritime emergencies in the area, “did not respond to any of the calls for assistance”, Sea-Eye said.
“There are now more than 800 people on the Sea-Eye 4, which has since set course for Lampedusa,” it said.
The migrants and 24 crew on board are facing an “unprecedented, highly stressful situation” and are “dependent on the rapid assignment of a port of safety”, it added.
The NGO SOS-Mediterranee said its Ocean Viking ship was also in the area and had provided food supplies to the Sea-Eye 4.
“Any delay by the authorities endangers the health and lives of the rescued people and our crew,” Sea-Eye chairman Gorden Isler was quoted as saying.
“It is shameful how Malta repeatedly shirks its responsibility and ignores distress calls.”
Italy is one of the main points of entry into Europe for migrants from North Africa, mainly from Libya and Tunisia, with tens of thousands of people seeking to cross the central Mediterranean each year.
The port of Lampedusa became a flashpoint for migration disputes under hardline former Italian interior minister Matteo Salvini.
Salvini went on trial last month for allegedly illegally blocking 147 rescued migrants from disembarking from a rescue ship and holding them in dire conditions.