Yes it will. I chatted with an official person on the Samsung Website and she said that the Verizon Galaxy Tab 7.7 is listed to be upgraded to ICS. She cannot confirm a date of when that will be released from Verizon.
The Verizon ICS OTA rollout started yesterday. There seems to be a wide spread problem with the update failing. XDA developers are debugging the issue but at this point the vast majority of people reporting their experience are reporting failure.
The update seems to require build FD01b in order to update properly however before debugging the install nobody on xda had heard of or seen that version. The current version being shipped on new units is version FD01.
It looks like Verizon or Samsung needs to step in to fix the problem. Any bets on how long that takes?
I gave up waiting and did it manually last month. I was in a Verizon store a few days ago and saw that the display 7.7 had an update notification. Tried to run it but it failed. Looks like it was the small update again. Did the OTA come in two pieces?