Yes, you must have default gateway in order to access the net outside the local net. Run the route command to see what gateways are assigned to which interfaces and for which destination addresses (usually 192.168.1.* will have gateway * because they are 1 hop from that machine and do not require an interface; you should see * addresses having gateway 192.168.1.1 for interface eth0 so that any non-local requests go through that router and out). Do you have a router with an inside ip of 192.168.1.1? If not, setting the gateway address to it will not have any effect.
You should follow the instructions given to configure both the .10 and .20 fedora machines to have their respective addresses and the subnet mask and gateway you have listed, using ifcfg-eth0. I am confused about you saying a fedora/redhat system should have hostname=router... is it acting as the router for the network (should have two network interfaces)?
To test that your network is working, you should ping the other computers on the local network, and then computers on the Internet (like www.google.com
, aol.com, microsoft.com). If you are not connected to the Internet at all, then default gateway has no effect on a tiny lan with 3 computers connected by a switch with static IP addresses - the ARP table in the switch will accomplish all the traffic routing you need.