The Socialist party (PSOE) lost ground in Sunday's elections in the northern regions of Galicia and the Basque Country to new anti-austerity party Podemos, which is seeking to replace it as Spain's main party on the left.
The PSOE finished fourth in the Basque region, behind Podemos, and was tied in Galicia with the En Marea coalition which includes Podemos with 14 seats each.
Rajoy's conservative Popular Party (PP) renewed its absolute majority in Galicia and lost just one seat in the Basque Country.
Pablo Casado, a senior PP member, said Socialist leader Pedro Sanchez "needs to reflect" and allow Rajoy to form a new government and end a months-long political impasse.
But Sanchez told a news conference that his opposition to a PP-led minority government had not changed.
"The 'no' to Mr Rajoy... has never been more justified," he said in his first public comments since the regional polls.
Spain is being run by a government without full powers after inconclusive elections in December and June that saw the PP win without an absolute majority and other parties fail to forge a rival coalition.
Rajoy's PP is six seats short of the absolute majority of 176 seats it needs in a parliamentary confidence vote, even with the support of centrist party Ciudadanos, and one extra seat from a minor Canary Islands party.
The Socialists scored their worst ever showing in both elections, finishing second.
An abstention by the party would be enough to enable a PP-led minority government under Rajoy, in power since 2011.
Parliament must usher in a new government by October 31 otherwise new elections will take place, around Christmas.
Rajoy did not directly address Sanchez on Monday, saying only that all political forces needed to "show responsibility" to end the lengthy political impasse which he dubbed a "pitiful show".
The PP "will continue to try to form a government because it is its obligation," he added.
Sanchez voted against a Rajoy-led government in a parliamentary vote of confidence earlier this month.
He opposes austerity measures imposed by the PP and argues that allegations of illegal financing and graft that have dogged the conservative party under Rajoy's watch have undermined his credibility.
Sanchez reiterated his wish to negotiate a "government of change" with arch-rival Podemos and centrists Ciudadanos.
But this strategy is complicated by the fact that Ciudadanos has steadfastly refused to be part of an alliance with Podemos, in large part due to differences over economic policy.
Sanchez is unpopular among many regional party leaders, or "barons", who think the party should help put an end to Spain's nine-month deadlock by admitting defeat, allowing a right-wing coalition government led by acting Prime Minister Mariano Rajoy to come to power.
Sanchez acknowledged there was a "deep debate" within the party over which path to take and announced he would seek backing for his stance a party leaders meeting on Saturday.
"I think it is very important that the Socialist party speak with a single voice," he said.
Sanchez also announced he would call for a leadership race on October 23rd to renew his position at the helm of the party.
"Whoever thinks they can do better should throw in their hat," he said.
By Michaela Cancela-Kieffer / AFP