What you've described is not the Prime Directive. In the Star Trek
world, the Federation
commonly interferes with the actions of other civilizations--many of the episodes of both the old and new shows deal with the efforts of the Enterprise
crew to influence or control the actions of other groups.
What the Prime Directive really says is that the Federation should not interfere with cultures who are so technologically behind their own that the culture shock would necessarily damage the other.
This results in a very different set of philosophical conclusions than you've drawn. The Prime Directive is not an endorsement of moral relativism at all--if the Federation decides that another group is doing something bad, they don't hesitate to get involved.
The Prime Directive is based in the notion that when two cultures of widely different technological levels meet, the weaker culture is likely to be completely overwhelmed by the stronger, resulting in a sort of cultural genocide. The evils of war, disease, and so on are indeed real, but they're lesser evils than the cultural annihilation that would result if the Federation intervened.
I wonder about the historical analysis you provide. In the case of Czechoslovakia, after all, the people made a pretty vocal protest against the occupying Soviet troops in 1968--and they got rolled over by tanks (see Tiananmen Square for a similar situation). If Gorbachev had decided to send out the tanks again, then the Velvet Revolution would have ended in failure as well. Free people really do get oppressed by minorities who use violence to get their way, and the "collapse" of the Soviet Union doesn't disprove that.