The petition was launched by the Austrian Medical Association (OeAeK) after the government scrapped a ban that was meant to come into force in May.
The ban was cancelled at the request of the far-right Freedom Party of Austria (FPOe), which entered government as a junior coalition partner after elections late last year.
The medical association's president Thomas Szekeres said the petition had got off to a "sensational start".
The petition proved so popular that registration of signatures on the interior ministry website had to be stopped for two hours on Friday because the ministry's servers were overloaded.
"This is a big vote and it has to make politicians rethink the issue," Szekeres said.
"We will keep collecting signatures and expressions of support so as to keep increasing the pressure," he went on.
Chancellor Sebastian Kurz, a non-smoker, is head of the center-right People's Party (OeVP), which had supported a ban while participating in the previous coalition government.
But FPOe Vice-Chancellor Heinz-Christian Strache, himself a smoker, insisted that the ban be dropped as a condition of entering government, saying it impinged on "freedom of choice".
Thirteen years after attempts to ban smoking began, Austria remains one of the last European countries where smoking in bars and restaurants is still permitted.
In theory smokers have to be seated in a separate smoking area -- although this is not always rigidly implemented -- but no separate area is necessary in establishments smaller than 50 square meters if the owner is happy to allow smoking on the premises.
Thirty percent of Austrians over the age of 15 are smokers, according to figures published by Eurostat, the third highest level in the European Union.
Around 13,000 Austrians die each year from smoking-related causes.