In short, no, the G1 cannot be used with Verizon at this time.
Currently the G1 operates on GSM networks via Sim cards (another such carrier would be AT&T) while Verizon uses the CDMA network for their phones (No sim cards). There are 'CSIM' or 'R-uim' cards used in some foreign countries which allow for use in GSM or CDMA networks, but I have not heard of them being widespread (if used at all) in the US.