No. Having solar panels does not make your household carbon neutral.
Imagine a two person household in Indianapolis. Without more qualifying information, their total carbon footprint is about 50 mtCO2e annually. Only 5% of that is from electricity (source).
Getting 100% of electricity from renewable sources (e.g. solar) is important and a great first step. Even if they did, however, their approximate carbon footprint would still be around 47 mtCO2e annually: