No.
All spellcasting focuses require the use of a free hand to manipulate it, unless you have a feature indicating otherwise. See the following rules from the PHB/SRD in the Spellcasting chapter.
A spellcaster must have a hand free to access a spell's material components--or to hold a spellcasting focus...
The best you could achieve without special features is to pull the eye out, use it in your hand as a spellcasting focus, and then reinsert the eye (or I suppose just fondle the eye while it's still in the socket). However, that's only if the DM even allows the eye to count as an arcane focus. Since that's not part of its item description, it would be up to the DM to rule on the availability of that option.