The singer is keen to focus on his music project with Alison Krauss and doesn't want to spoil the band's "legacy" with more reunion dates.
The legendary '70s rockers reformed for a one-off gig at London's O2 arena in December.
Bandmates Jimmy Page and John Paul Jones are reportedly keen to extend the reunion and believe the band still has "something to offer".
A source told the Sunday Mirror: "Despite the enormous offer, the decision did not come down to money. They always said they would do the one-off show and then see how they felt.
"Jimmy had enjoyed the concert in December enough to want to tour. He argued they still had something to offer. He likes the idea of another chapter in the band - the grown-up tour.
"John sided with Jimmy. He loved making music with the others again. But Robert wanted to leave last year's concert as their legacy. They had proved they could still do it and that was enough.
"He has other commitments and is happier looking forward to those. Robert put the mockers on the tour."