I have done this to the bathroom door, not out of choice, but because I had to cut a hole in the door to get our son out of the bathroom!! A drawer was open and there was no other way in!!
Anyway, what I did was square out the hole that I had cut, and then ripped down 2x2 lumber that would fit between the two 1/8" panels on all 4 sides. This gave stability to the door. On these ripped pieces, I routered a channel around the edge 1/4" wide (maybe it was 1/8", can't remember) and the depth of the tempered glass. I installed those ripped pieces into the opening so that routered edge was available for the glass to fit into. Then I used clear caulk on that edge, and pressed the glass into it, cleaning all the extra off. I used some moulding around the window which nailed into those 2x pieces and gave a little extra ease of mind to the glass. I also used the moulding on the outside of the door.
I have no problem with stability on the door, and although not professional quality, I'm quite proud of it...
Oh, my door is 26" x 78" and I only put in an 18"x16" piece of glass in the upper section, not a piece that fits a majority of the door...