The Texas senator’s tweet, posted on Saturday, was in response to what many Republicans see as “nannying” from scientific advisors.
The public health experts - with the strong support of the Democrats - are urging people to cancel their travel plans and stay at home over Thanksgiving, in a bid to stop the spread of the virus.
Mr Cruz posted an image of a Thanksgiving turkey, with the caption: “Come and get it”.
His tweet was “liked” over 90,000 times and retweeted almost 30,000 times.
But, coming amid a worsening Covid crisis in his state, many were not amused.
Texas has almost 1.1 million cases of Covid confirmed, and over 20,000 deaths.
On the day that Mr Cruz tweeted, at least 79 new coronavirus deaths were registered, and 8,534 new cases were reported in his state.
Over the past week, there has been an average of 11,759 cases per day - an increase of 40 per cent from the average two weeks earlier.
El Paso is the current epicentre of the pandemic, and on the day Mr Cruz tweeted the Texas division of emergency management announced that a team of 36 National Guard troops had been deployed to "provide mortuary affairs support".
One man responded to Mr Cruz’s tweet with a photo of a body on a stretcher.
“More than 255,000 Americans have died from coronavirus and a sitting U.S. senator is pretending someone, anyone wants to take away his turkey,” tweeted Kyle Griffiths, a producer with MSNBC.
Another man, Isaac Armendariz, tweeted: “My Dad is in ICU in El Paso right now with Covid and these idiots don’t care to them it’s just numbers.”
Others were circulating a 22 July interview with Mr Cruz, in which he suggested that the pandemic was simply a ploy to damage Donald Trump’s electoral chances.
"If it ends up that Biden wins in November - I hope he doesn’t, I don’t think he will - but if he does, I guarantee you, the week after the election suddenly all those Democratic governors, all those Democratic mayors, will say: everything is magically better. Go back to work, go back to school, suddenly the problems are sold.
"You won’t even have to wait for Biden to be sworn in. All they need is election day, and suddenly their willingness to just destroy peoples’ lives and livelihoods - they will have accomplished their task.
“That’s wrong. It’s cynical. And we shouldn’t be a part of it.”
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