I have an Italian pine tree's roots breaking up my sidewalk, can this be fixed without killing the tree??

yes it can .

there are a couple of things you can and need to do

first of all the reason your sidewalk is breaking up or being lifted is mainly because you probably never deep root water your tree , meaning the water the tree uses to feed upon is only near the top of the ground , so its roots grow at the top of the ground causing your problems ,

you need to use a deep root feeding pipe that connects to your hose to water deeper in the ground

now with that said ,you can breakup the existing sidewalk and re- pour it over its current roots . or you can cut the root so as to stop the root from growing any bigger , doing more damage to your walkway