The reason you water chemical fertilizer after you apply it is to
1. wash it off the grass blades,
2. dissolve it at least a little bit, and
3. let it into the soil at the root zone.

That does not take much water; however, if you do not wash it off the grass blades, then the next morning when the dew hits, the fertilizer will dissolve into the dew (which is almost no water) forming a very strong salt solution on the blades which ultimately kills the grass. If you apply chemical fert in front of too much moisture, then it will dissolve and wash through the soil right past the roots.

You should be able to apply the day after the rain, water lightly, and have a good app.