UPDATE: The iPad 2 will be produced in a CDMA version compatible with the Verizon Wireless network. This model hits stores on March 11, 2011.
UPDATE Jan 11, 2011: Verizon Communications CFO Francis Shammo announced today that Apple will manufacture iPads that will connect directly to the Verizon network. Recently the company has been offering Wi-Fi iPads in promotional bundles with MiFi portable hotspot units.
Rumor has it that Verizon and Apple are in talks to provide service for a future model of the iPad. Currently by all accounts the iPad hardware will not even support the Verizon network, so the answer is no.
Future iPad models could support the current Verizon network, or more likely Apple will wait until Verizon upgrades to a newer network (LTE) and then AT&T and Verizon will be using the same technology. In this case (not until at least 2011) one new iPad will work on both networks.