The captain of the Open Arms, Marc Reig, sent a letter on Monday to Spain's embassy in Malta asking that Madrid grant asylum to the minors, saying they all "fulfill the conditions for recognition as refugees".
Another 150 migrants who were plucked from several boats in the Mediterranean are on board the Open Arms, off the coast of Italy's southern Lampedusa island.
Both Malta and Italy have refused permission for the boat to dock and unload all the migrants.
Asked if Madrid would agree to Reig's request, Public Works Minister Jose Luis Abalos told private television Telecinco the Open Arms captain did not have the "legal competence or authority" to demand asylum for the minors.
Under Spanish law asylum applications must be lodged in person or by a legally accredited representative.
The minister also recalled that under international agreements, rescued migrants should be taken to the closest available port which in the case of the Open Arms is in Italy.
Abalos defended Spain's record in rescuing migrants at sea, saying the coast guard had saved 50,000 migrants last year, echoing recent comments made by other Spanish ministers.
"It is not fair to question the government of Spain, or the reputation of the government of Spain, when it comes to the issue of rescues," he added.
Proactiva Open Arms, which operates the Open Arms rescue ship, has called for European countries to agree to take in migrants it has picked up.
Italy's far-right Interior Minister Matteo Salvini has insisted since coming to power last year that rescued migrants can land in Italy only if an agreement is already in place with other European countries to look after them.
Oscar-winning Spanish actor Javier Bardem on Monday called on Spanish Prime Minister Pedro Sanchez to take a leadership role within the European Union for the distribution of migrants Open Arms has rescued, joining other celebrities including Antonio Banderas and Richard Gere who have urged governments to act.
"We think that Spain is the ideal and right country to do so since it's the country of origin of the NGO Open Arms, which is doing a necessary and extraordinary job for human dignity and to save the lives of people who are fleeing from situations that we can't even begin to imagine," Bardem said in a video posted on Twitter.