Actually, dual citizenship was illegal until 1967, and it is still illegal (just ignored by the State Department) for those gaining US citizenship through naturalization (legally, they have to renounce their other citizenship).
It is true that it is mainly a state department issue as far as recognizing dual citizenship. However, there have been several rulings by the Supreme Court, which ultimately decides whether laws are Constitutional, and my explanation describes one of the ways in which dual citizenship is obtained, and cannot be revoked.
Now, if an American born citizen goes to Israel, and obtains Israel citizenship without renouncing American citizenship, what's illegal about it? What could be illegal about it? I'm afraid there is no recourse for those who wish to end this practice. The USSC has basically ruled that the only way to lose American citizenship is to specifically and unambiguously renounce it, in writing, in the proper forum (i.e. inside an embassy or state dept office in front of a U.S. official).