Experts say Threads could attract Twitter users unhappy with recent changes to the platform.
Threads - which is not launching in the European Union for now - allows users to post up to 500 characters, and has many features similar to Twitter.
The app passed five million sign-ups in its first four hours, Mr Zuckerberg said.
Earlier, he said that keeping the platform "friendly... will ultimately be the key to its success".
But Twitter chief Elon Musk responded: "It is infinitely preferable to be attacked by strangers on Twitter, than indulge in the false happiness of hide-the-pain Instagram."
When asked on Threads whether the app will be "bigger than Twitter", Mr Zuckerberg said: "It'll take some time, but I think there should be a public conversations app with 1 billion+ people on it.
"Twitter has had the opportunity to do this but hasn't nailed it. Hopefully we will."
Competitors have criticised the amount of data the app might use. This may include health, financial, and browsing data linked to users' identities, according to the Apple App Store.
Threads is now available to download in over 100 countries including the UK, but not yet in the EU because of regulatory concerns.
-- Courtesy of BBC News