The reason there is no D in refrigerator is because it's not needed to make it sound correct.
However with fridge without the d would be frige which would not sound right.
So the D is added to Fridge to make it sound correct and have a more correct spelling.
The other answer below is a good insight as to why fridge has a D and refrigerator has no D.
Why does FRIDGE have a D?
It’s an attempt to make the spelling connote the correct pronunciation:
FRIG would rhyme with RIG and is, also, used as a curse.
FRIGE would have a long I sound.
FRIDGE looks like RIDGE with an F added, and it rhymes with RIDGE.
2. Why doesn’t refrigerate have a D?
It comes from Latin refrigeratus and, in Latin, I can be short or long (no need for a silent E) and in Latin, G is soft when it comes before an E.