Macron and Conte also vowed to respect worldwide maritime laws related to humanitarian aid of stranded migrants.
In addition, Italian minister of foreign affairs Enzo Moavero Milanese summoned the French ambassador on Wednesday morning to discuss the matter.
Speaking to the senate Salvini said "he hoped for an official apology as soon as possible", from France.
Italy has received both criticism and praise for turning away the Aquarius rescue ship over the weekend as it made its way across the Mediterranean Sea carrying 629 migrants.
French government spokesman Benjamin Griveaux quoted President Emmanuel Macron as saying there was "a degree of cynicism and irresponsibility in the Italian government's behavior", the news agency reported.
French President Emmanuel Macron harshly criticized the decision, angering Italy.
Italy's rejection of a migrant ship is shameful but the responsibility lies with Europe as a whole, United Nations refugee chief Filippo Grandi told Reuters on Wednesday.
The refusal of Italy and Malta to take in the rescue ship Aquarius packed with migrants this week has shone a light on the flaws of European solidarity and underlined the urgent need to reform the Dublin asylum system.
A U.S. Navy vessel is waiting to learn where it can disembark 40 migrants who survived a capsizing off Libya.
"It's very clear that Europe needs to reform its asylum system in a collective manner; there's a lot of resistance to that but there's no other way", Grandi said.
More than 200 on board were minors.
"We believe an axis of the willing is needed to fight illegal migration", he said.
Italy had demanded an apology after the French president accused its new populist Italian government of irresponsible behavior for refusing to allow a rescue ship carrying 629 migrants to dock at an Italian port.
"Italy and France must deepen their bilateral and European cooperation to implement an effective migration policy with countries of origin and transit", the statement said.
The Aquarius is headed to Valencia in Spain after Italy shut its ports to hundreds of migrants onboard earlier in the week.
Salvini has accused charities that rescue migrants of working with human traffickers but said Italy would not stop rescuing migrant boats itself.
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