Well, I guess I shouldn't have stated it that strongly. It depends somewhat on where you draw the line between homophones and multiple definitions of the same word. But the idea that Japanese has an abnormally large number of homophones really has no support from any actual research (which should be very easy to do considering it's just counting up things).
In order for homophones to *require* the use of kanji, the number of homophones compared to English would have to be much, much larger. It defies all logic and reason that there could be that many homophones in a language that cannot be disambiguated by context.
People are fond of looking in dictionaries and coming up with lists like this:
高工 【こうこう】 (abbr) higher technical school
孝行 【こうこう】 (adj-na,n) filial piety, (P)
後攻 【こうこう】 (n) (baseball) taking the field first, thus batting second
黄口 【こうこう】 (n) baby chicken, young and inexperienced person
後項 【こうこう】 (n) following or later or last clause or article, etc.
港口 【こうこう】 (n) harbor entrance
膏肓 【こうこう】 (n) incurable disease
口腔 【こうこう】 (n) mouth cavity
香香 【こうこう】 (n) pickled vegetables
坑口 【こうこう】 (n) pithead, minehead
公侯 【こうこう】 (n) princes and marquises, great feudal lords
高校 【こうこう】 (n) senior high school, (P)
交 【こうこう】 (n) sexual union
鉱坑 【こうこう】 (n) shaft, mine, pit
皇考 【こうこう】 (n) the (deceased) father of the current emperor
航行 【こうこう】 (n,vs) cruise
洸洸 【こうこう】 valiant, brave, surge (of water)
They look at a list like this and conclude that you must need kanji to tell these apart. The problem with this approach is twofold:
1) a dictionary lists many words, most of which are not in common use.
2) Even with a long list of homophones such as these, it would be very hard for a literate native speaker to be confused about these in an actual contextual passage, just because the definitions are so different.
I've said this before, but learners are often drawn to the homophone explanation because kanji are a great help to someone trying to translate/interpret something that is beyond their level of language -- when you lack the ability to figure out the words from contextual information, having the kanji is a big help (assuming you have some easy way to look up the kanji words, like WWWJDIC).
Unfortunately this myth is fueled by native speakers as well -- they know that they find romaji or all kana harder to read than normal text, but they're not really sure why, so they just guess and come up with the homophone thing because that's what everyone else says.
Look up "sit" in an English dictionary; mine lists over 100 definitions. Do we need 100 different kanji for that word?