If Israel did not exist, they would still hate us. It's much simpler to blame everyone else for your problems rather than deal with them on your own. It's easier for many governments in the Arab world to blame the infidel for their incompetence than it is for them to actually fix matters or take the blame.
I'm not "blaming" Israel. I'm saying they hate Israel and all her friends. I've no doubt they'd come up with other reasons, but THAT is #1. (IMO)
Note 1: 'Because we support Israel' is bogus - they see Israel as a western/european construct, that's sufficient to earn their hatred:
if Zionism had been a Lutheran movement, they'd hate it only somewhat less.
Note 2: only somewhat does not equate the two, it only means that the conflict would not go away.