This will be your lucky day - you get to tell your husband he is dead wrong! :)
In fact, riding a stationary bike is excellent for those who have problems with weight-bearing activities (i.e. your feet). All you need to do is switch up the speed/level at which you pedal to work in intervals and it can be a great fat burning/cardio activity.
Google spinning. Here is one link that I found: http://bicycling.about.com/od/trainingandfitness/gr/spinn...
You may be able to find spinning aps for your iPod/iPhone or other helpful links in order to maybe facilitate some sort of spinning routine on your own. The key is to always keep moving and work in intervals (either pedal faster for like 60 seconds, put more resistance on for 60 seconds, etc. and do this for a total of 30-60 minutes). You don't have to do a spinning workout if you don't want to; I just suggested that if you want some sort of formal program to follow. I took a spin class once (on a regular spin bike, yes there is a difference) and it hurt my butt and my um, "bike seat/lady area" really badly!
Yes, there are other activities out there that will help you drop the weight faster but that doesn't mean that using a stationary bike would not be effective.
Ride the bike, do the work, burn the calories, melt the fat, put on a hot little outfit, and then tell him "I told you so - haha!"
Best wishes - you totally can do this!