1. “Republican” means a member of the US Republican party. That’s the only real meaning of the word. It’s a team in the political game. In absolutely no way is it a well-defined meaningful general political philosophy, even if you pick one particular day in history. The political positions of that party (and the Democratic party, and practically every political party anywhere) changes over time, and in the case of US Republicans, quite quickly. People point out for example that the positions of Ronald Reagan were “more to the left” (though that too is of highly dubious real meaning) than Obama, whom ridiculously high numbers of Americans think might be a foreign agent , an evil Muslim, or even the Anti-Christ . Republican President Eisenhower (and former World War II 5-star general) would no doubt have deplored most of the current Republican positions, not to mention President Lincoln. So the question doesn’t seem well enough defined to really be answerable, except as no that doesn’t make much sense.
2. If we take the question as “in the current general compass of US politics in 2015, would Kirk’s sentiments be sort of like the current Republican Party”, I’d tend to say one reason not is that Kirk tends to mostly be intelligent and make sense, while the main thing I notice about current Republicans is that they tend to be sort of bat-shit crazy, suggesting that most scientists are conspiring to invent global climate change, for instance.
3. Now, Kirk was often a pretty hard-core bad-ass starship commander who was almost always out to assert the Federation’s “foreign policy” and regulations. But even in the original Star Trek, the Federation was astronomically more “liberal” that current US Republicans. The Prime Directive, for example, puts non-interference with less developed populations above the survival of Star Fleet units. As in, you’re supposed to let yourself die before you even let natives see that you have space-age technology, so as not to interfere with their development. (of course, if you mean the Kirk of JJ Abrams’ films, then THAT Kirk pisses all over that because he’s an immature narcissistic snot that TOS-unviverse Kirk (and TOS-universe Star Fleet) would never let command an away team, let alone a starship, but that just makes JJ Kirk a loose cannon, not a Republican.) And if you want to see how much tolerance Kirk has for entrepreneurs, see how he treats Harry Mudd or the tribble merchant.
4. On the other hand, Kirk really didn’t appreciate that time space hippies hitched a ride on his ship. On the other hand, he tried to be patient and the guy in charge had brain problems that made him a jerk and there was that whole hijacking issue . Still that just makes Kirk a non-space-hippie ship captain, not a Republican.