If you want to call High Intensity Interval Training (HIIT), or Cross-Fit, “Islamic Exercise”, or come up with some other Arabic terms, feel free. If you want to change Kung-Fu or Muay Thai to something like “Islamic Martial Arts”, because you think it somehow diminishes us to use imported terms, go ahead. But it is silly; just as it is silly to object to the term “Democracy”. But if you want to coin a different term, you are free to do so.
Democracy has many extent varieties, in both theory and practice, and there is no reason why we could not develop our own model to better suit our societies and culture, and to hopefully alleviate the flaws present in existing models. But at the end of the day, when you are talking about a form of government that incorporates the participation of the general public in the selection of the government and the management of the affairs of the society, that is going to fall under the political category of a democratic system.; though, again, you can come up with any name for it that satisfies your cultural insecurity.
The arguments against Democracy tend to change whenever you confront them. You prove that democratic processes operate under the rule of law, which for an Islamic government would mean, the rule of the Shari’ah; thus the argument that Democracy means giving absolute sovereignty to the people is entirely fallacious. So they try to come up with other arguments; “the people should not be allowed to vote on establishing an Islamic Constitution, because that recognizes their right to potentially refuse Shari’ah; that is kufr”. I don’t really agree with the argument, but it is easily dealt with anyway. Simply let a party run for office stating that they will draft an Islamic constitution if they win; if they win there is no need for a constitutional referendum. Simple. Next argument?
Finally we are always left with two things: 1.) Democracy is a western concept and it is foreign to Islam, and 2.) The West will never allow us to have Democracy.
The first point is irrelevant. We do not have a political system in Islam; we have laws, and we have the command to engage in Shura, and the system for implementing these things is adaptable. There is no reason whatsoever why we should not incorporate useful ideas from any source if they are beneficial and do not interfere with religion. The second point is not in any way an argument against Democracy, so there is no need to address it in this context.
What you are left with then, are merely critiques of flaws or manipulation of democratic mechanisms in specific places; which, of course, do not invalidate Democracy as a theory or as a system. That something is not “perfect” or that it does not inaugurate a Utopia, is not a reasonable criticism. But, as you see, the arguments, one by one, can be easily pulverized, and by the end of the discussion, we are very far away from the initial lunacy of saying “Democracy is Kufr”.