Bautista Agut had played two singles matches for Spain at this week's tournament in Madrid, losing to Andrey Rublev before defeating Croatia's Nikola Mektic.
However, Spain will be without the 29 year-old – its number two player behind world number-one Rafael Nadal – for the rest of the competition.
"We want to accompany Roberto in these hard times. He is an exceptional tennis player who symbolises the highest values of tennis and sports," Royal Spanish Tennis Federation (RFET) president Miguel Diaz said in a statement. "He is an example, not only as an athlete, but also as a person. I want to transfer in my name all the support of Spanish tennis."
Five-time Davis Cup champion Spain faces Argentina in the quarter-finals.