Rakesh Agrawal was hired by the company about two months ago as the director of strategy.
The company said in a Twitter post on Saturday that Agrawal is no longer with the company, which has zero tolerance towards offensive language and behaviour.
"Rakesh Agrawal is no longer with the company. Treat everyone with respect. No excuses. PayPal has zero tolerance," the company tweeted.
Agrawal sent a series of offensive and sometimes incoherent Twitter messages while he was at a jazz festival in New Orleans.
Some of the tweets were directed at PayPal's vice president of communications Christina Smedley, with Agrawal calling her "useless middle manager" and using an expletive for her.
Another tweet, which was incoherent, read, "People who should be fire from paypal Don Christmas a pool a kick (sic)."
He went on to tweet that he could make Foursquare a 50 billion dollar company.
"I can turn foursquare into a USD 50 bill OK n company let's chat," he tweeted.
After several incoherent and grammatically incorrect posts, Agrawal tweeted that he had quit the firm.
"Oh. I quit pay a tonight because of self at son and aortic and ah our and hill e a s th (sic)," he tweeted.
He then quickly deleted the offending tweets and publicly apologised to PayPal president David Marcus and vice president of growth and global strategy Stan Chudnovsky.
Agrawal claimed he was trying to send the messages to a colleague as he was trying out his new phone and that he did not realise the messages were being tweeted instead of being sent to his colleague.
"Last night I was using a new phone that I bought because I wanted to test experiences on android. Those messages were meant for a colleague," he said in a later tweet.