My system has 2 network cards.

I want to configure eth0 for the system IP address, 192.168.2.2
I want to add another IP address to eth0:1 for an oracle database, 192.168.2.3
I want the second network card to run backups, eth1, IP address 192.168.2.4

The problem I am having is the routing table is not working correctly. If I pull the network connection on eth1, all network traffic stops. I think the problem can be seen in the routing table below, however I don't know how to fix it. I am trying to find the correct configuration in the redhat files to make the correct routing table.

Here are my configuration files:

/etc/sysconfig/network
----------------------------
NETWORKING=yes
HOSTNAME=system2
GATEWAY=192.168.2.1
GATEWAYDEV=eth0

/etc/sysconfig/network-scripts/ifcfg-eth0
---------------------------------------------
DEVICE=eth0
ONBOOT=yes
BOOTPROTO=static
IPADDR=192.168.2.2
NETMASK=255.255.255.0
TYPE=Ethernet
USERCTL=no
NETWORK=137.230.2.0
BROADCAST=137.230.2.255
PEERDNS=no

/etc/sysconfig/network-scripts/ifcfg-eth0:1
----------------------------------------------
DEVICE=eth0:1
ONBOOT=yes
BOOTPROTO=static
IPADDR=137.230.2.3
NETMASK=255.255.255.0
TYPE=Ethernet
USERCTL=no
NETWORK=137.230.2.0
BROADCAST=137.230.2.255
PEERDNS=no

/etc/sysconfig/network-scripts/ifcfg-eth1
---------------------------------------------
DEVICE=eth1
ONBOOT=yes
BOOTPROTO=static
IPADDR=137.230.2.4
NETMASK=255.255.255.0
TYPE=Ethernet
USERCTL=no
NETWORK=137.230.2.0
BROADCAST=137.230.2.255
PEERDNS=no

output from netstat -rn
-------------------------
Kernel IP routing table
Destination Gateway Genmask Flags MSS Window irtt Iface
137.230.2.0 0.0.0.0 255.255.255.0 U 40 0 0 eth1
137.230.2.0 0.0.0.0 255.255.255.0 U 40 0 0 eth1
127.0.0.0 0.0.0.0 255.0.0.0 U 40 0 0 lo
0.0.0.0 137.230.2.1 0.0.0.0 UG 40 0 0 eth0

Your config of eth1 looks correct (this is you public interface).
Your eth0 and eth0:1 look like the have some issues.
I don't think you need eth0:1.
eth0 (private interface)
Have a subnet matching the address/netmask
IPADDRESS=192.168.0.1
NETMASK=255.255.255.0
SUBNET=192.168.0.0
You don't need to create another address for oracle it uses port 1521 by default.
When you start NETCA just give it the ip address of 192.168.0.1 and port 1521.
Note it will not be accessable from the internet. If you want it to be you need to add the public address to.
Again DON'T bind two address you don't need to, and make sure you subnets match your address. You can use netcfg to do this.
If you have any trouble with oracle I can help you. I am an Oracle DBA.
Good luck.

I am not trying to create the linux box as a router. Both network interfaces are public. I am trying to run one network card with oracle and have the second network for all the backup traffic. That way I am not using bandwidth on the oracle network card during backups. We typically create an IP address for each Oracle database running on a system. That way we can move the database to another system at any time without changing the hostname or IP on the operating system. This also allows us to create clusters where databases can fail to other systems. This system actually has two oracle databases running on it. Eth0 actually is running with 3 IP's. I only asked the question about two because I replicated the same problem on a server without oracle and I did not want to make the question too complicated.

The problem is that you cannot have two physical network interfaces from the same box on the same network--at least, not using the standard route command. Perhaps the ip command has extended features that would allow this.

When I say public I mean public IP.
Then 192.168.0.0/16 subnet is a private IP subnet ( not routable over the internet).
The 137.230.2.3 is public (routable over the internet) assigned by you ISP.

Assigning many IPs to a network card does not aid in bandwidth utilization. You would need additional network cards. Which I don't suggest. It can create routing issues when servers are connect to multiple subnets. Networks should be connected with routers and servers whould exist on a give subnet. If you need more bandwidth use gigabit NICs ans switches.

OK back to the progblem.
What is the subnet/subnet mask and default route for your network. Is it
SUBNET=192.168.0.0
MASK=255.255.255.0
GATEWAY=192.168.0.1

What is the output of
/sbin/ifconfig
dmesg | grep net