Welcome to the group and it sounds like your thermo couplin is no good or not in the proper position. To explain, this tube has to heat up and this is what will activate your gas valve to open, you can try this. First look into where the pilot flame is, see if the slender tube looking thingy, IE thermo couple, is where the pilot flame is going to hit it and heat it up when your holding the button in. If so. it is probably not working and it is easy and inexpensive to replace. Another way to test it is to use a
propane torch to heat the tube and see if this will open the gas valve to keep the
fridge lit.