Robinson was suspended from international cricket pending an investigation after his Twitter posts resurfaced when he played against New Zealand at Lord's in the first Test last week.
The 27-year-old, the pick of England's bowlers in the drawn game, released a statement apologising for his "thoughtless and irresponsible" tweets.
Sussex, who described the posts as "completely unacceptable", said Robinson would miss Vitality Blast Twenty20 games against Gloucestershire and Hampshire Hawks on Friday and Saturday, with the player set to spend time with his family.
"Player and staff welfare -- including mental health and wellbeing -- is a priority for the club and, as such, Sussex Cricket fully supports Ollie in his decision," the team said in a statement.
"When he is ready to return, Ollie will be welcomed back to the club, where discussions with the remainder of the squad over the last week have made clear that he retains the support of all his Sussex team mates."
The England and Wales Cricket Board (ECB) said on Tuesday it would take "relevant and appropriate action" after offensive historical Twitter posts from other England players began circulating on social media.
Robinson was replaced by Olly Stone for England's second Test against New Zealand at Edgbaston, which began earlier on Thursday.