Parag Agrawal, the company's chief executive, says that Musk won't join the board.
The appointment of Mr Musk was due to take effect on Saturday after he revealed last week that he had bought a stake in the social media platform.
However, Elon shared that same morning that he wouldn't be joining the board.
He said that the firm will remain open to his input.
Just over an hour after the announcement, Mr Musk cryptically sent out a single message.
Over the weekend, Mr Musk suggested changes to the premium subscription service, including banning advertising and giving the option to pay in dogecoin.
Mr Musk asked his more than 81m followers whether or not the headquarters of the social network should be turned into a homeless shelter.
He created a poll asking if the letter should be removed from the social network, with only two options: yes or no.
The board members had to act in the best interests of the company and all shareholders, as the company had believed it was the best path forward, with Mr Musk being offered a seat on the board.
Addressing Mr Musk, Mr Agrawal said: "I believe this is for the best."
He said that they value input from shareholders whether they are on the board or not.
Elon is our biggest shareholder and we will remain open to his input.
The company's largest shareholder is Mr Musk, who has a stake of more than four times that of Jack Dorsey.
After Mr Musk revealed his stake in the company, shares in the company soared.
The announcement that the chief executive ofTesla had become the largest shareholder in the company and that he was invited to join the board caused a lot of unhappiness among the employees.
The company was worried about what impact he would have on the moderation of content in the future.