The 930-mile journey across the Mediterranean from Sicily to Valencia took almost a week.
Salvini said he plans to meet this week with Pope Francis, who has dedicated much of his pontificate to urging countries to welcome and integrate migrants.
David Noguera tells The Associated Press that "I have mixed feelings".
The Italian coast guard vessel Dattilo carrying migrants approaches the eastern port of Valencia, Spain, Sunday, Jun. 17, 2018.
On Valencia's quayside, 1,000 red cross workers were on hand to greet the migrants as they stepped off the ship.
Rome's decision put the migrant influx back to the fore ahead of a European Union summit on June 28-29, where leaders are supposed to hammer out an overhaul of the bloc's asylum rules.
"They are very shaken", Suarez said.
Initial medical examinations show no serious injuries or illnesses among the refugees. Two other boats, the Aquarius and another Italian ship, the Orione, are due to arrive later Sunday.
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"These people could not be left adrift in the Mediterranean, where they would face death", she told Spanish radio Cope.
The Aquarius originally picked up about 630 migrants - including pregnant women and children - from six rubber dinghies in the Mediterranean Sea off Libya.
"Although the rescue and transfers of the 630 people were initiated and coordinated by the Italian Maritime Rescue Coordination Centre (MRCC), the Italian authorities denied Aquarius authorisation to bring them ashore in the closest port of safety in Italy".
"They also look exhausted", she told AFP. A private aid ship said it couldn't assume the unexpected passengers from Trenton because Italy wasn't assigning it a port within its navigational capacity.
The destination wasn't immediately announced, but the coast guard said it wouldn't dock until at least Tuesday.
Interior Minister Matteo Salvini said Monday that the government intends to hold a special sort of "census" of Roma people in Italy. Malta also sent it away.
The Aquarius, operated by Doctors Without Borders and SOS Mediterranee Sea, and another Italian government boat with migrants will dock in Valencia in the next few hours.
The first of three vessels entered Harbour soon after dawn. The insistence of the new Italian government, led by the League and the Five Star Movement, to curb the arrival of migrants and asylum seekers has already rocked relations with France, which on Saturday offered to help Spain take the migrants. Country leaders are expected to discuss the policies surrounding the issue at their European Council meeting June 28-29.
The warmer weather has caused a spike in migration from North Africa to Europe.