But in fact, as I've pointed out before, this gets things precisely backwards. Only objectivists can admit their own fallibility. Relativism would make everyone infallible, for whatever you may believe, the sheer fact of your belief in it suffices to make it "true for you". (What drivel!)
Consistent objectivists are cautious about imposing their views on others. They should always have the nagging concern, "what if I'm mistaken?" The objectivist recognizes that not all opinions are equal, so perhaps he has something to learn from others. Perhaps they are right and he is wrong. Given this possibility, he really must take care to hear others' views, investigate the evidence carefully, and come to a well-reasoned conclusion.
The relativist can recognize no such thing. He takes his moral views to be infallible. If he wants to impose his arbitrary prejudices on others, then he couldn't possibly be mistaken in doing so. Caution is unnecessary; reflection, superfluous. The relativist has no reason to refrain from authoritarianism.
Even if the objectivist is confident in their views, they need not impose them on others. I think that most people would have better lives if they watched less TV, and spent their leisure time on more worthwhile activities (like philosophy!). But I wouldn't dream of imposing this view on others. Indeed, I think it would be objectively wrong for me to do so! Liberal tolerance is not just a "compromise" I make for the sake of ensuring that others don't try to interfere with me. I think it's positively normative, a good thing in and of itself. Respect for other people entails letting them make their own decisions, and even their own mistakes.
The objectivist can hold, as I do, that tolerance is an objective value. That intolerance is positively wrong. That if I were intolerant, that would be bad of me. Again, the relativist cannot recognize any of this. Perhaps he happens to value tolerance, or perhaps he doesn't. Neither option is an improvement on the other. Even if he doesn't value tolerance for its own sake, he might behave tolerantly for prudential reasons, as required by the social "peace treaty". But if such a person could get away with breaking the treaty, he would have no reason not to. This relativist is tolerant only insofar as he is weak. Once powerful enough to impose all his preferences on others without fear of reprisal, he has no reason to hesitate. He doesn't recognize the concerns or choices of others as having any intrinsic import; perhaps he will choose to respect others, but he might just as well choose not to. After all, he's a relativist. "Anything goes."
Finally, let me address a specific remark from the Doc:
[T]he moral relativist is more open to differences in moral behavior, as long as they aren't excessively offensive to his sensibilities. For example, the relativist may find abortion to be distasteful, but will not coerce a woman to come to term against her will. As long as his wife or daughter aren't forced to have abortions, the abortions of others do not represent a threat in the form of a challenge to an overarching principle.
But that's nonsense. If the relativist is "excessively offended" by abortion, as many conservatives are, then he will try to advance his preferences -- possibly without regard for the value of women's autonomy, if the latter is not something he personally cares about. Replace "abortion" with "infanticide" or "murder" in the above quote (as many conservatives consider them morally equivalent). Do you really think that people will just ignore murder so long as it doesn't affect them? If so, you seem to be describing an amoralist, not a relativist that has any sincere moral convictions.
Conversely, consider the moral objectivist who has doubts about the morality of abortion. For reasons discussed above, he may be tentative about his own fallibility here, and so reluctant to force his (possibly mistaken) views on others. He doesn't want to make a mistake, after all. (This is something relativists never worry about.) And even if confident in this case, he must recognize the autonomy of women as having some moral weight. Finally, if he is utterly convinced that abortion is murder, then we can expect him to do all he can to oppose it -- perhaps by force. But of course the same is true of the relativist who holds such personal convictions.
The only reason for thinking that relativism would lead one to be more "liberal" here is if relativism leads you to simply not care about others. Objectivism can lead us to the principle of "live and let live". Relativism goes further, to "live and let die".