It depends on whether you're assuming the traditional kaiju immunity to all conventional weapons.
According to the opening bit of Pacific Rim, those CAN be killed with immense effort and tremendous quantities of conventional weapons, which is actually better than usual - Japanese kaiju are usually just flatly immune to anything except other kaiju, kaiju-scale mecha and stuff, or one-shot prototype superweapons. And even then they usually don't actually die, just go dormant for a couple of years (with occasional exceptions, mostly in the older movies, like the original Godzilla and the Oxygen Destroyer ... and even that spawned a later kaiju).
In some movies Godzilla can "feed off" radiation, so even nukes may not be entirely reliable.
---
I agree kaiju require materials stronger than conventional bone, cartilage and muscle, but the numbers I've seen suggest that you don't need supermaterials on the carbon-nanotube-for-space-elevators scale. The bigger problems are:
1) ground support on anything but hard rock - this has been one of the big problems of "super tank" attempts IIRC
2) heat dispersal
3) food sources
How thick are Godzilla's legs? Showa Godzilla is 50 meters tall, which probably makes him about 2500-3000 metric tons, given that he's built much like a human but heavier (bulkier, plus tail) - a 1.8 meter, 90kg human scaled up to that size would be about 2000 metric tons. If his legs at the narrowest part are ... say 3 meters thick, and circular, that means an area of about 7 square meters. Thus, Godzilla's weight of about 25 megaNewtons is distributed over 14 square meters (at the narrowest point) = 1.78 megapascals. That's only the standing force, multiply by maybe 2-3 for walking, and 2-3 more for safety factor ... say 10 MPa total. Not a materials scientist, how excessive is that?
Heisei Godzilla is 100 meters tall. That makes him eight times the mass = 20,000 - 24,000 metric tons, with legs... maybe 6 meters thick, with a total standing area of 56.5 square meters. 200 megaNewtons over 56.5 square meters = 3.54 megapascals, so allowing for walking and safety factor ... about 20 MPa.
The interesting thing is, since the leg area scales up as the square while mass scales up as the cube, the pressure put on the legs increases only linearly (less, if you make your legs more-than-proportionally thicker). That's why I don't think kaiju really need space elevator materials.
(Anguirus is the size of Showa Godzilla and a quadruped. He might actually be workable with conventional biological materials, assuming extreme use of lightening via air sacs and stuff, and very thick legs.)
