Yes, with this clarification: "culture" is too broad a term. It includes all the humans of a group, and their entire way of life: families, food, music, homes, land, law, literature, etc. etc., the good and the bad, all mixed together. We don't make war on cultures. We make war on something more limited: the aggressors and their military assets.
The problem with islam is that every moslem is an aggressor. The koran commands it. The only way to end the war is to end islam. the war ends when the last moslem dies.
That's also true. The absolute prohibition is: you do not, in a directly intended manner, slay noncombatants or--- another way to put it --- those who are not deliberate accomplices in acts of aggression.
Every moslem supports islam. Every moslem supports jihad. They must if they are to follow the koran.
The Bible calls this "shedding innocent blood" and the Lord declares that He "hates" this, and says it is an "abomination to My soul".
Is a person who supports the political/religious system that is designed and commanded to fight against, slave and kill all people's not under that system innocent? I don't think so.
"Is it moral to eradicate a culture that attacks you?...In fact, it is immoral not to eradicate them."
Once again, you are required to use physical force, even lethal force, to stop a determined physical aggressor. As for an aggressor culture, it's more complex. If the culture has hell-bent features, you can attack those features via other means, for example by legal suppression of its financers and enablers, by argument, education and media campaigns, by exposing its elements of ugliness, evil, and futility, or by appealing to conversion to a better standard of love and reason.
This is true. After pacification of the more aggressive components you can eradicate the culture by a long program of supression and conversion. That is, every copy of the koran must be destroyed, every mosque must be razed, no moslem should be allowed to practice islam. All should be converted to Christianity and once their children or grandchildren are proven to have no knowledge of the koran then when that first generation of converts dies off, the war will be over. Unfortunately a moslem can never be trusted even if he says he converted since islam allows them, nay commands them, to lie if they can gain advantage from it.
The war ends with the death of the last moslem.
In my estimation, Muhammad was a delusional charlatan, a lecher, and a man of blood, as recorded in the hadiths. Follow any path in the writings by him or about him, and they all veer off into perversity.
It's an exceedingly dangerous situation, because you have people with a transcendant belief in a hell-bent system. However, "kill them all" is a hell-bent system in itself. I caution against becoming what you are fighting,
Here's the Wiki version of it, and here's the music video, which I think is pretty good. The English subtitles were done in the UK.
I know what some people will say ("Taqiyyah!") but you can't discount the fact that the Muslim world is divided and in turmoil about violent jihad, some for it and some against.
"For" and "Against" are still opposites, a difference that has been written in blood, and it is unjust to fail to distinguish between the two.