"Over just the past few weeks, the daily number of cases have gone from an average of about 2,000, to more than 5,000," Mr Abbott said on Sunday.
Several southern and western states have recorded a surge in cases after lockdown restrictions were eased.
The number of reported infections in the US has now surpassed 2.5 million.
Over 125,000 Covid-19 patients are confirmed to have died nationwide - more than in any other country.