There's a BBC article up on how many angry responses on Twitter followed a single reference to "the Persian Gulf." It also mentions that a reference to the Persian Gulf by that name recently prompted some Arab diplomats to walk out of a NATO meeting. For reasons of national or ethnic pride, many across the Arab world refer to it as "the Arabian Gulf," and insist that others do the same, despite a 2006 ruling by a UN panel of experts. To avoid stirring up a wasp's nest, National Geographic lists both names on its maps, and the BBC just calls it "the Gulf."
A similar situation exists with the Sea of Japan, which is a name that incenses many Chinese and South Koreans. They insist that the name which should be used in the English Language is "the East Sea." North Korea, in contrast, prefers "the East Sea of Korea." The matter of renaming it has been raised with both the International Hydrographic Institute and the United Nations.
I will note that, when I was learning Geography in school, both "the Persian Gulf" and "the Sea of Japan" were taught as the standard English names for those bodies of water. I don't see any need to change those terms. Let "the Arabian Gulf" (or its equivalent in the Arabian language) remain the standard term for the Persian Gulf in the Arabian language. Let "the East Sea" (or its equivalents in the relevant languages) remain the name of the Sea of Japan in the Chinese or Korean languages. All the fight is over the terms used in English, and that should be something that only English speakers will decide.
For inspiration, look at another potentially politicized name for a body of water which, somehow, has remained beyond any political controversy. Can you imagine a French person referring to the water between France and Britain as "the English Channel"? Never! It is as much French as English! So the name in French is "La Manche" (the Sleeve) and the name in English is "the English Channel," and no one calls foul on the other's choice. London is Londres in French, Moskva is Moscow in English, Germany is Deutschland in German, and so it goes. Surely, that's the civilized solution.