The vessel is a separate plane of existence so you cannot see through your familiar's eyes
The Bottles Respite feature states:
[...] The interior of the vessel is an extradimensional space [...]
The following question establishes that extradimensional spaces are separate planes of existence, and thus, entering the vessel prevents you from seeing through your familiar's eyes:
Even if you could, spells still require a clear path to their target
The rules on "Targets" states:
[...] To target something, you must have a clear path to it, so it can't be behind total cover. [...]
Meanwhile, the flesh to stone spell has you choosing a creature to be affected (it has you targeting a creature) and as such you are required to have a clear path to that creature, which you cannot have from within the vessel.
Now, if there is a spell that requires you to see something, but does not require you to target that something, then yes, you could use it while seeing through your familiar's eyes. I cannot think of particularly many spells that function like this, though one such example would be misty step which does not require a clear path to its destination because it has a range of self.
A small note, as @MivaScott pointed out in the comments: the find familiar spell states (emphasis mine):
[...] Additionally, as an action, you can see through your familiar's eyes and hear what it hears until the start of your next turn, gaining the benefits of any special senses that the familiar has. [...]
Beginning to see through your familiar's senses requires you to spend an action, so you could not start to look through them and cast an action spell on the same turn (in most cases... there are exceptions such as through the Quickened Spell Metamagic and Action Surge features). Though one way around this is simply to start seeing through your familiar's senses at least a turn (six seconds) before you would want to case whatever spell you are trying to cast.