no it does not.
both feet have to be in contact at the start of the pitch. The stride foot obviously leaves the rubber and the only requirement is that the back foot if it does leave the rubber is that it must stay in contact with the ground, or level with the ground if there is a hole, and can't replant and repush can only "drag"
both feet leaving the ground is leaping which is illegal
replanting is referred to as crow hopping and is also illegal
At the same time prior to ball release. (added for clarity)
A pointless rule IMO - by the way widely broken if you look close enough. If they dumped the leaping rule I think the rest of the pitching rules might be more uniformly obeyed and called. But I digress.
The one I often heard preaching about the illegality of the back foot leaving the rubber before release was Bill Redmer and certainly he had his own fairly unique ideas.
As the others said, no - back foot does not have to be in contact with rubber on release. I've always wondered why people think that the back foot needs to stay in contact. Have you ever tried to pitch like that? Very difficult! ...Not to mention horrible mechanically.