Trump triggered an international uproar when he made his comments in response to last week's deadly shootings in California by two Muslims who authorities said were radicalised.
"You are a disgrace not only to the GOP but to all America," Prince Alwaleed, the chairman of Kingdom Holding 4280.SE , said on his Twitter account, addressing Trump and referring to the Republican Party.
"Withdraw from the US presidential race as you will never win," the prince added.
Within hours, Trump's response came back, also on Twitter.
"Dopey Prince @Alwaleed_Talal wants to control our US politicians with daddy’s money," he said. "Can’t do it when I get elected."
Trump's comments have already cost him business in the Middle East, with a major chain of department stores halting sales of his glitzy "Trump Home" line of lamps, mirrors and jewellery boxes.
On Thursday, Dubai real estate firm Damac, which is building a $6 billion golf complex with Trump, stripped the property of his name and image.
Prince Alwaleed, a nephew of Saudi Arabia's King Salman, has holdings in a number of international companies, including Twitter and Citigroup. In July he said he will donate $32 billion to charity in coming years via Alwaleed Philanthropies.
