CelebHeights peg him at 5'8". That sounds about right. Back when Downey was out of prison and out of work about a dozen years ago, I used to see him at our sons' baseball and soccer games at the local park, and he's not tall at all.*
These days, Hollywood casts short leading men with leading ladies who are taller than them (e.g., Gwyneth Paltrow with Downey, Nicole Kidman with Tom Cruise).
In general, leading men are not as disproportionately tall as during Golden Age Hollywood (John Wayne, Jimmy Stewart, Gary Cooper, and Cary Grant were all close to a half foot taller than the average American man of their time). Overall, I'd say that the decline in bias in Hollywood toward tall role models is a good thing.
Height used to be a pretty good marker of having enjoyed good nurture (e.g, had plenty to eat as a child). The Tory cabinet of prime minister Lord Salisbury in 1895 averaged six feet at a time when that was about a half foot taller than the average British man. From a female husband-hunting perspective, evidence that a man's family provided well for him when he was a child is evidence of a lot of good things. There's no downside to growing up so that you attain close to your genetic maximum of height.
Over time though, the systematic nutritional and health deficits that prevent a youth from a lower class background of attaining the full height of which his genes are capable have diminished. The NBA is full of guys who grew up on welfare. (Although in Downey's case, the kind of heavy drug use from very early age might have knocked an inch off his height.)
So, height is increasingly a measure less of nurture and more of nature. And, as somebody who is 6'4", the genetic advantages and disadvantages of being unusually tall seem like a mixed bag. If people weren't somewhat subjectively biased in favor of tall men like myself, I'd probably say the objective tradeoffs (clumsiness, head-banging, etc.) aren't really worth it. The human body isn't optimized for my height.
So, the continuing prejudice in favor of the tall seems increasingly pointless because it's now mostly a nature difference masquerading as a nurture difference, and there's no terribly good reason to want genes for additional height to be favored. Thus, the fact that Hollywood role models currently come in all heights seems, on the whole, like a good thing.
* By the way, don't get the impression from this that Downey is some kind of regular guy. I said hello to him and he said hello back, very friendly, but the Charisma Gap was astounding. In a social setting that was blase about minor levels of celebrity -- e.g., the baseball team mom was an Emmy-nominated character actress -- Downey, in disgrace, was the cynosure of all eyes of team parents. Just lounging on the grass watching his kid take infield practice, he's magnetic.
You know those scenes in Iron Man where Tony Stark wakes up from a horrible dream? I suspect Downey's Method Acting technique for this is to tell himself: "I had a nightmare that I had to move back to the Valley!"