In a video message posted to Facebook on Monday afternoon, Mr Morrison said targets were not practical as COVID "writes its own rules".
"You don't get to set the agenda," he said.
"You have to be able to respond quickly to when things change and we've had to deal with a lot of changes.
"Rather than set targets that can get knocked about by every to and fro of international supply chains and other disruptions that can occur, we are just getting on with it."
Launching the first of a series of daily vaccination data updates to be published online, he said Australia's rate of 1.2 million to date was comparable to other major countries.
Chief Medical Officer Paul Kelly said the updated AstraZeneca advice had forced a recalibration of vaccine targets.
"I won't give a number or date but we absolutely committed to providing the vaccine to anyone, any adult Australian, who wants the vaccine as quickly as possible," he told reporters in Canberra.
Part of the recalibration will include changing where vaccines are available to better suit the new plan.
Professor Kelly said the initial phases of vaccinating quarantine, border, health, aged care and disability workers and residents were on track to be completed mid-year.
He said the rare side effect linked to the AstraZeneca showed there was some advantage to not being at the top of world rankings on vaccine rollout speed.
National COVID-19 Commissioner Jane Halton called for calm after the ditched timetable threw the rollout into chaos.
"The trick now is for people just to calm down a little bit and get back to basics," she told Nine on Monday.
"We need to vaccinate the nation, we need to have the vaccines to do that, we're going to get Pfizer at the end of this year and there will be 40 million doses in total of Pfizer."
Opposition Leader Anthony Albanese described the rollout as a shambles.
"This is an absolute failure by the federal government. It was failing before the issue with AstraZeneca," he said.
Cabinet minister David Littleproud defended the vaccine strategy, arguing Australia's progress should not be compared to other countries.
"I don't think any Australian would want the Chinese vaccine or the Sputnik vaccine," he told Nine.
"Australia has been calm and methodical about making sure that we give the best vaccine with confidence, and however long it takes, it takes."
Epidemiologist Mary-Louise McLaws said unless national vaccination rates ramp up to between 100,000 and 120,000 per day, it will take two years for Australians to be fully vaccinated.
On Sunday just over 12,000 jabs were administered.
The McKell Institute has found vaccine delays will increase the chance of lockdowns and hurt the economy, warning the cost could be more than $1.4 billion.
Trade Minister Dan Tehan will travel to Europe on Wednesday to urge his German, Belgian and French counterparts to do what they can to increase vaccine production.
Many of the world's vaccines, including AstraZeneca and Pfizer, are manufactured in Europe.
But because of the continent's export controls, it has effectively blocked contracted supplies to countries including Australia.