Rajoy said Spain had shown solidarity with Greece as part of the euro zone by helping with its bailout and urged Greece to fulfill its obligations and keep its promises.
"We are not responsible for the frustration generated by the radical Greek left that promised the Greeks something it couldn't deliver on," he said.
The Greek government on Sunday sought to play down the row, saying Tsipras was obliged to tell Greeks how the negotiations went and that his comments had been misinterpreted.
"The new Greek government does not categorize European countries and people as friends or enemies," a government official told Reuters. "Any misinterpretation of the Greek prime minister's speech does not help the dialogue."
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Spain, Portugal and Ireland have faced stringent austerity programs and structural reforms in return for European assistance, in Spain's case for its troubled banking system.
Tsipras rejected criticism that Athens had staged a climbdown to secure an extension of its financial lifeline from the euro zone, saying anger among German conservatives showed that his government had won concessions.
Spain's anti-establishment Podemos movement has topped some opinion polls, making it a serious threat to Rajoy's conservative People's Party in an election which must be held by the end of this year.
Portugal will also have elections after the summer but no anti-austerity force as potent as Syriza or Podemos has so far emerged there.
Ireland will hold an election in 2016. Opinion polls show the left-wing Sinn Fein has become the closest rival to Irish Prime Minister Enda Kenny's center-right Fine Gael party.
In an interview published before Tsipras made his speech, Prime Minister Pedro Passos Coelho denied that Portugal had taken a hard line in negotiations on the Greek deal at the Eurogroup of euro zone finance ministers in February.
Passos Coelho aligned himself with euro zone governments which have called for policies to promote economic growth but without trying to walk away from austerity as in Greece.
Portugal had to take its own bailout in 2011 but left the program last year. Finance Minister Maria Luis Albuquerque said on Saturday Lisbon would start repaying its loans to the IMF next month, giving back 6 billion euros.
Greece remains in its EU/IMF program, almost five years and two bailouts after it had to seek international help.
Tsipras has portrayed the Eurogroup deal as a victory for Greece, even though it meant extending the bailout program he had promised voters to scrap.
So far he has public backing. A poll conducted by the University of Macedonia for SKAI TV showed 56 percent of Greeks believed the extension had been a success, compared with 24 percent who said it represented a failure.