* France's Macron says he didn't mean to offend Italy
* Italian PM, Macron confirm Friday lunch meeting
* Pope says countries must change mindset on migration
* Aquarius hits rough seas on way to Spain, migrants suffer
* U.S. ship abandoned migrant bodies at sea - report (Adds Conte comment, statement from U.S. Navy)
By Steve Scherer and Massimiliano Di Giorgio
ROME, June 14 (Reuters) - Italy and France tried to patch up a row over immigration on Thursday as Pope Francis urged politicians everywhere to work together on helping refugees and respect their dignity.
Italy summoned France's envoy on Wednesday and demanded an apology from President Emmanuel Macron, who had said Rome's move to block a migrant rescue ship from its ports was an act of "cynicism and irresponsibility".
Macron, in a phone call late on Wednesday with Italian Prime Minister Giuseppe Conte, said he did not mean to offend "Italy and the Italian people", according to a statement.
The two leaders confirmed a lunch meeting on Friday to discuss "new initiatives" on immigration, a day after Italian Interior Minister Matteo Salvini announced an "axis" with Germany and Austria to fight illegal migration.
"France does not want escalation; it's counter productive. We need to maintain dialogue," a source close to Macron said as the president visited the western town of Rochefort. However, the source added that Macron was not "taking back anything".
Italy's Conte later said on Facebook he would take to Paris "Italy's request for broad collaboration and solidarity on immigration at a European level". Conte will meet German Chancellor Angela Merkel in Berlin on Monday.
Salvini has vowed to continue to block foreign humanitarian boats from Italian ports as Europe wrestles with how to share the responsibility of handling migrants trying to enter the EU from war zones and poor countries, largely across Africa and the Middle East.
More than 1.8 million migrants have arrived in Europe since 2014, and Italy is now sheltering more than 170,000 asylum seekers, as well as an estimated 500,000 unregistered migrants. A European Union summit will discuss the bloc's asylum rules at the end of the month.
Pope Francis, who has made the defence of refugees a plank of his papacy, rebuked politicians for not respecting the dignity of migrants and demanded "a change in mindset".
Speaking at a conference on migration at the Vatican, the pope said countries must work together and "move from considering others as threats to our comfort to valuing them as persons whose life experience and values can contribute greatly to the enrichment of our society".
The row centred on the charity ship Aquarius, which both Italy and Malta refused to let dock at their ports. It was carrying 629 migrants and is heading to Spain, which has offered safe harbour, escorted by two Italian ships.
On Thursday some 80 migrants suffered from sea sickness due to rough conditions, according to Doctors without Borders, one of the charities that runs the Aquarius.
Ropes have been strung around the ship to make it easier to walk on deck, video from the Aquarius showed. Among the migrants are women and small children who are not expected to disembark before Saturday evening. They were rescued off the coast of Libya last week.
Despite criticism from the pope and the U.N. refugee agency, which called the treatment of the migrants on the Aquarius "shameful", Salvini has not changed his position.
"Ships belonging to foreign organisations and flying foreign flags cannot dictate Italy's immigration policy," Salvini, who also heads the anti-immigrant League party, said on Wednesday.
A U.S. Navy ship recovered 40 survivors after a shipwreck on Tuesday, but it still had not disembarked them on Thursday. In a statement, the sixth fleet said the U.S. ship, the Trenton, was "coordinating" with authorities.
A Dutch-flagged humanitarian ship, the Sea Watch 3, had offered to take them on board, but not without a written guarantee from the coastguard that it will be allowed to dock in an Italian port, which it did not receive.
The U.S. Navy denied a report on Italy's la Repubblica website that said the Trenton had been forced to put 12 bodies it recovered after the shipwreck back in the sea because it did not have a refrigerated storage for them.
Instead, the Trenton crew had spotted "approximately" 12 bodies in the water, but they were not recovered, according to the statement.
(Additional reporting by Philip Pullella in VATICAN CITY, Michel Rose in PARIS and Marine Pennetier in ROCHEFORT, France; editing by David Stamp and Alison Williams)
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