"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.
The virus has now been linked to more than 500,000 deaths worldwide, according to data compiled by Johns Hopkins University in the US.
The spike has led officials in Texas, Florida and other states to tighten restrictions on business again, with warnings that hospitals may soon be overwhelmed.
On Sunday, Mr Abbott said that as many as 5,000 people a day were being admitted to hospital for treatment.
US Vice-President Mike Pence said that Texas would be issued additional Covid-19 testing kits for as long was necessary.
"We are going to make sure that Texas, and your healthcare system in Texas, have the resources, have the supplies, have the personnel to meet this moment," Mr Pence said.
The vice-president also urged Texans to wear masks "wherever its indicated", saying "we know from experience, it will slow the spread of the coronavirus".