The semi-autonomous region of Catalonia, which has its own language and culture, held a referendum on independence on October 1.
Rajoy had earlier stressed that Puigdemont's response would be crucial in deciding "events over the coming days" and he said he "just needs to say he didn't declare independence".
The Spanish government regarded the vote as illegal and attempted to prevent it with a violent police crackdown.
The deadline sets the clock ticking on Spain's most serious political emergency since its return to democracy four decades ago.
Spain today gave Catalonia's separatist leader until next week to clarify whether he intends to push ahead with independence, warning that Madrid would take control of the region if it insisted on breaking away.
Puigdemont issued a symbolic declaration of independence from Spain on Tuesday night but then immediately suspended it and called for negotiations with the Madrid government.
"There is no possible mediation between democratic law and disobedience and unlawfulness", Rajoy said, throwing the ball back to the Barcelona-based Catalan authorities for the next move. This requirement is prior and obliged to implementation of any of measures that Executive may adopt under article 155 of Constitution. An Interior Ministry official later apologized for the injuries but laid the blame on the Catalan government for having encouraged people to vote.
Marta Rivas, a regional lawmaker with the Catalonia Si Que es Pot anti-establishment party, warned that applying Article 155 to curb the region's autonomy could backfire and produce more protests.
About 2.3 million Catalans - or 43 percent of the electorate in the northeastern region - voted in the independence referendum. Rajoy's government had repeatedly refused to grant Catalonia permission to hold a referendum on the grounds it was unconstitutional, since it would only poll a portion of Spain's 46 million residents.
The leader of the opposition Socialist Party, Pedro Sanchez, backed Rajoy's demand for clarification in order to "get out of the swamp".
"If an autonomous community does not comply obligations that Constitution or or laws impose on it, or act in such a way as to seriously adhere to general interest of Spain, Government, upon request to President of Autonomous Community and, in case If not attended, with approval by an absolute majority of Senate, it will be able to take necessary measures to oblige that one to forced fulfilment of se obligations or for protection of aforementioned general interest".
The idea, Sánchez added, was to find a formula to "allow for Catalonia to remain a part of Spain".
Lauren Frayer, reporting for NPR from Barcelona, spoke to graduate student Eugenio Juliá, who "is paying close attention because he's anxious the terms of his grad school scholarship might change if Catalonia leaves Spain and the European Union", Frayer says. "I didn't know what he was saying".
"It was not very straightforward".