Murray was aiming to reach his second quarterfinal in as many weeks as he continues to recover from hip problems.
Rudd beat Murray 7-5, 6-4 and then paid tribute to his comeback.
“On the other side of the net there was a legend of the game over the past decade, even what he’s been doing now for a year, fighting to recover from a very serious injury has been an inspiration for me, ”Rudd said.
The opening set was watched at 5-5 before Rudd stepped up to take the lead.
Murray then fell behind early in the second set, losing five straight games in a streak that included a streak of seven straight unforced errors.
There was good news for Cameron Norrie, who reached the next round after beating fellow Briton Dan Evans 7-6 (7-3), 6-3.
Murray has been given a wildcard for the Indian Wells main draw, which begins Monday, and he is set to make his first appearance there since 2017.