The group said it "urgently requests a swift resolution and a designated port of safety". "This could compromise all the work and cooperation between us over the past few years".
Salvini and Italian Transportation Minister Danilo Toninelli, who is part of the 5-Star Movement faction in the new government, said in a joint statement Sunday that it was Malta's responsibility to "open its ports for the hundreds of the rescued on the NGO ship Aquarius".
A humanitarian ship with more than 600 migrants on board remains adrift in global waters as Italy and Malta are at loggerheads over which country should take them.
SOS Mediterranee said the passengers on its ship, the Aquarius, included 400 people who were picked up by the Italian navy, the country's coastguard and private cargo ships and transferred.
The decision by Italian interior minister Matteo Salvini, who is also head of the far-right League party, represents an opening gambit to make good on his electoral promises to halt the flow of migrants into the country.
Maltese Premier Joseph Muscat also refused to take in the Aquarius ship of the humanitarian group SOS Mediterranee.
Tweeting on Sunday, Salvini said: "Starting today, Italy, too, begins to say NO to the trafficking of human beings, NO to the business of clandestine immigration".
As of Monday morning, the Aquarius was circling in global waters off Malta after the Italian maritime coordination centre in Rome ordered the vessel to stay put (see map).
The French organisation said that among those saved and brought on board are 123 unaccompanied minors, 11 small children and seven pregnant women.
Mr Salvini says he wants to "stop the death boats".
Under global law, people rescued at sea should be brought to the nearest and safest port for assistance.
The Maltese authorities have been insisting that since the rescue happened in the Libyan search and rescue area and was coordinated from Rome, Malta is neither the coordinating authority nor has the competence to settle this issue, thus it would not take in the migrants and asylum seekers.
"My aim is to guarantee a peaceful life for these youths in Africa and for our children in Italy", he said, using the Twitter hashtag "We are shutting the ports".
One Aquarius crew member, Alessandro Porro, told Italian news channel Sky TG24 that they desperately needed to know which port to go to.
Farther west in the Mediterranean, Spain's maritime rescue service saved 334 migrants and recovered four bodies from boats it intercepted trying to reach Europe over the weekend. The migrants were taken to a naval base in Tripoli. The lawlessness in Libya has made it a popular place for migrants to try to depart for Europe.
The United Nations estimates at least 500 people have died in 2018 trying to cross the central Mediterranean, following more than 2,800 fatalities a year ago.