Sweden applied to join Nato after Russia invaded Ukraine - but needs Turkey, already a member, to approve.
Kurdish protesters in Sweden hung an effigy of Mr Erdogan this month, followed by the Quran burning.
"Sweden should not expect support from us for Nato," Erdogan said in response.
"It is clear that those who caused such a disgrace in front of our country's embassy can no longer expect any benevolence from us regarding their application."
Saturday's protest - but not the burning of the book itself - was given prior approval by Swedish authorities.
Erdogan condemned the latest protest, carried out by a far-right politician from a Danish party, as blasphemy not to be defended by free speech.
The Swedish governments also criticised the protest.
"Sweden has a far-reaching freedom of expression, but it does not imply that the Swedish government, or myself, support the opinions expressed," Swedish Foreign Minister Tobias Billstrom said on Saturday.
Responding to Mr Erdogan's remarks on Monday, Mr Billstrom said that he wanted to understand exactly what the Turkish leader said before commenting.
"Sweden will respect the agreement that exists between Sweden, Finland and Turkey regarding our Nato membership," he added.
Sweden, along with Finland, applied to join Nato after Russia invaded Ukraine, but the recent protests have heightened tensions.
Nato's secretary general Jens Stoltenberg said that freedom of expression was a "precious commodity" in Nato countries, and that these acts, while inappropriate, were not "automatically illegal".
Turkey, a majority Muslim country, denounced the Swedish government's decision to allow the protest as "completely unacceptable".
"No one has the right to humiliate the saints," said Mr Erdogan in his televised remarks on Monday.
"When we say something, we say it honestly, and when someone dishonours us, we put them in their place."
Defence Minister Hulusi Akar said Turkey cancelled a visit by his Swedish counterpart Pal Jonson after "observing that no measures were taken over the... disgusting protests".
A flurry of earlier visits by Sweden's top ministers to Turkey's capital Ankara raised hopes the trip could ease objections to Sweden's accession.
As Turkey is already a Nato member, it can block another country from joining, and has made several demands of Sweden already. That includes the extradition of some Kurds that it claims are terrorists.
Earlier this month, the Swedish prime minister said Kurdish protesters in Stockholm who hung an effigy of Turkey's president from a lamppost were trying to sabotage Sweden's Nato application.
A Swedish minister branded the stunt as "deplorable", but Turkey said the condemnation was not enough.