In general, yes: you don't have to respond, even if you assume the question isn't rhetorical and the person is actually looking for an answer that they don't already know. Those answers are very easy to find and the major online resources do a decent job of debunking them.

But if you do decide that you really have to answer it, you need to be careful to clearly and systematically undermine each aspect of mis/disinformation that your response includes.

First, distance yourself from each incorrect belief: "they might think that...". I would even be inclined to add "but that doesn't make sense, because...". This addresses the faulty arguments.

Second, point out each individual factual inaccuracy as it's mentioned: "they might have been told that Earth has gone back and forth between heating up and cooling down in the last hundred years, but that's not true. Climate scientists have consistently been discussing global warming since at least the 1950s." This addresses the faulty factual claims behind those faulty arguments.

Science communicators and scientific skeptics have wrestled with this problem for decades: even the process of debunking an idea can help cement it in people's minds - and that's when it's clearly being debunked. Your intentions may have been good, but your comment didn't clearly distance the ideas and it made no effort at all to point out the inaccuracies buried under the argument.

Don't do anything because some random person online told you to - do things because you don't want to spread lies about the climate. I'm sorry that I was excessively terse. It seemed clear that all three of those statements were your opinions, especially since you said you were not going to speak for everyone who is conservative. That's a disclaimer that I only add before I present my opinions.