One Web Server Two NIC with Two Different Wan Network
Windows, and most OSes only "use" one default gateway at a time. It doesn't distinguish on a per NIC basis. If you have two default gateways in your routing table on the server (the default gateway in the NIC screen becomes a default route (0.0.0.0) in the routing table) then the one with the lowest metric will be used. You can use 'route print' to see that the two 0.0.0.0 entries have different metrics.
So, if you remove the default gateway from one of the NICs that has incoming traffic, then that traffic will be sent back out of the default gateway in the routing table (the default gateway on the other NIC) and dropped on the other side ( as unsolicited traffic ). This happens because there are no routes to the destination of the incoming traffic! You could set up local routing to send specific traffic to required interface, but you'd need to know the specific network ranges you wanted to deal with.
If you need redundancy for two ISPs, then you should have that redundancy set up on an actual router, or a switch that can handle it.