I don't think Archer was a racist. If he had been, I doubt SF would have even considered appointing him to captain the Enterprise, no matter who his father was.
The problem with Archer's attitude is that the scene that opens "Broken Bow" was supposed to set up the resentment of humans toward Vulcans because the Vulcans were throwing up roadblocks to slow humanity's effort to build a deep space starship.
I don't consider that to be much of an excuse to spit bile every time you see someone with pointed ears or threaten to "knock a 90 pound female Vulcan on her ass." Incidentally, I suspect that had he tried, he would have ended up in traction for three months.
I think that his resentment toward the Vulcans would have been better justified if the Vulcans had, for example, withheld treatment that could have prolonged his father's life, at least to see his Warp 5 engine fly, even if he never got to finish it.