Using Intel Galileo Ethernet Port
I suppose you mean the "Galileo integrated Ethernet port" (the on-board RJ45). Then the answer is "no, not directly". The whole code provided by Intel in Arduino 1.6 IDE applies to the W5100 based shield, not the integrated Ethernet interface (which is a bit stupid, imho...)
Technically, however, this is possible, but you will need to create a library in C/C++, which interfaces with the system lib (the Linux one). You can then access to the wonderful world of BSD sockets (much more powerful than then sockets within the W5100)
Note however that the configuration of the integrated Ethernet board is not very easy with the Galileo, because the default Yocto image does not allow you to change default IP parameters (the Yocto image uses DHCP by default). So technically, you need then to deal with a startup script which changes the IP parameters at startup, but it's pretty heavy to do.