Mr Rajoy accepted the mandate but warned he might fail if opposition parties again refused to back him.
His Popular Party (PP) lost its absolute majority in December`s election but remains the biggest party.
Most parties have declared they will not back him, including the Socialists under leader Pedro Sanchez.
"We want to change Rajoy`s government and this is why we will vote `no` in a confidence vote," Mr Sanchez said.
Mr Rajoy needs to form a minority or coalition government which must pass a vote of confidence in parliament.
Spain`s political turmoil began after the election on 20 December when voters, frustrated with economic crisis and austerity measures, broke the country`s traditional two-party dominance.
After the parties filed to agree on a new government, a new election was held on 26 June where, for a second time, no party won an absolute majority.
Spain election: Rajoy asked to form government by King Felipe
30 July 2016
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The King of Spain has asked acting Prime Minister Mariano Rajoy to try to form a government after the second inconclusive election in six months.
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