Yes you can. But you have to mod out the front fender to make the clearance. The easiest way is to fill the screw holes in the fender with jb weld or similliar. Then drill a new hole so the fender sits higher on the forks. I do not remeber the exact measurments, so you will need to measure. It should not be more than a few millimeters.
Then you will need to raise the fork tubes about 5 - 10 mm in the triple trees. I would start with 5mm and test it out.
The gixxer 600 comes with a 70 series front tire. I have never ridden with one on the r6. I have heard of people doing the swap though. It is supposed to give you a little more lean angle, and a more compliant ride.
The numbers 120/70/17 mean the tire is 120 mm wide, 70% of the width tall (84mm tall), and 17 inches for the rim size. So the 70 series tire is 72 mm tall. That is a 12mm difference in hieght.
The 60 tire is part of the reason why the front end is so sloppy and will add to your tank slapper. Of coarse, you probably won't notice a difference unless you really push your bike.
You'll need to raise the forks about 10mm's like dj says.
Seen it done and heard lots of great things about doing it, so i'm gonna this winter, along with a scott's steering damper
It will slow your turn-in, however, that's what raising your fork's is intended to get back for you. Their is an excellant thread going on this at r6messagenet.com in the technical section. Maybe I'll cut and paste here tomorrow when i have more time. Lot's of pro's and con's.
If you go to a 70 you might need to raise the fender a bit to allow for clearance. Depends sometimes on what brand tire you go to.
Also when you put the 70 on you might think there's enough clearance but you need to allow for firstly stones etc. to be able to get between tire & fender & also allow for the fact that at speed the fender pushes down closer to the tire than at standstill due to wind resistance.
I've heard of guys that have put a 70 on without raising the fender thinking they have enough clearance only to find out down the track that there fender is wearing away & is half the thickness it used to be.