A lawmaker aligned with the government proposed that the vote be postponed to November and that a new committee be set up to study details around the process of making such changes first.
The decision angered opposition parliamentarians and protesters, who had gathered outside the parliament to press their demands for constitutional change and for the removal of Prime Minister Prayuth Chan-ocha, a former junta leader.
Protesters say the current constitution was drafted to ensure he kept power after elections last year. Prayuth says the vote was fair.
The protests are the biggest challenge to the Thai military and palace-dominated establishment since Prayuth took power in a 2014 coup. The biggest protest drew tens of thousands of people at the weekend.
Some protesters say the constitution also gives too much power to King Maha Vajiralongkorn and have called for his influence to be reduced, challenging a longstanding taboo of not criticizing the monarchy.
The 2017 constitution was written by a military-appointed committee and passed a nationwide referendum in 2016 at which opposition campaigning was banned.
The assembly combines an elected lower House of Representatives and a military-appointed Senate.