The couple - who married in a private ceremony in Los Angeles earlier this month - are said to have scrapped the trip in favour of spending time alone in their California home.
"They've chosen to stay at home. They were thinking of going away but decided to stay where they're most comfortable - in the house with their dogs," a source told The Sun.
"They want to squeeze in a proper honeymoon later in the year once the hype about their wedding has died down. Robbie also didn't want to run the risk of them being photographed on their romantic getaway either."
Williams recently invited his new mother-in-law to move into the couple's home as she struggled to meet mortgage payments.