No, it does not damage the chip. Sorry for the confusion. It just means that the Ethernet controller will not work without the correct crystal.
In any case, even with the 32.768kHz crystal soldered on X1, it shouldn’t cause the controller to not go past the ‘Connect to Network’ screen. From your picture, everything looks fine and I can’t see any obvious issue (other than X1).
The fact that the LEDs on the Ethernet jack do not light up (assuming you have connected it to your router) is a strong indication that the Ethernet controller cannot initialize for some reason. One possible reason could be that some signal line is being pulled to ground. I suggest that you go through all the solder joints, and check if any solder joint is accidentally touching the ground plane around it (the grid area is the ground plane, it spans the entire PCB). When in doubt, use your soldering iron to melt the solder joint (i.e. reflow the solder joint) so that the joint will retract from the ground plane.