DoorSafety.com is focused more on doors used in commercial and industrial buildings, but I am glad to answer your questions. 

If I understand your questions correctly, you are concerned about the possibility of breaking the glass when you attempt to remove it from the door for cleaning. 

It's difficult to give you direction without seeing the type of glass and the framing that secures it to the door. Usually, the framing is removable; some frames have snap-in moldings and other frames use screws to attach it to the door. Most likely, the frame is sealed to the door, at least on the outside face of the door, which makes taking the glass out more difficult. If the frame is sealed to the glass or door, you might need to use a heat gun to soften the sealant as you remove the frame. If the moldings are plastic, you'll need to be careful so that the heat gun does not melt or otherwise distort the moldings. 

As you described the glass itself, it might be that the two panes of glass are sealed together as a unit (i.e., insulated glass) with a space between them. If that is the case, you will not be able to get inside them to clean them without breaking the seal.

Your best option might be to contact a local glass and glazing company and have them inspect the door for their recommendations.

Speaking from personal experience, the seals in the insulated glass of several of our double-hung windows failed, causing the glass to appear permanently "foggy" between the pieces of glass. Our local glass company was able to make new insulated panels that fit our windows. It wasn't cheap, but it was less expensive than replacing the windows entirely.  It's all relative.

Another option you might have is to find a completely new window "kit" for your door that has a new frame and glass. Changing out the original glass and replacing it with new, might be the best option.

Good luck!