"The most important human right is the right to live"
No, that would be one of the pseudo "human rights" made out and pushed by the EUropean state funded "humanists", such as the right to housing [at the expense of others].
I just doesn't make any sense in terms of natural rights - although yeah, it sounds cool and caring on paper - because living is not something that can be granted and/or justified by law in the first place (well, at least not by human laws that is. But surely, you weren't talking about God?).
While "human rights" do indeed apply only to the living (As far as I'm informed, nobody ever started a campaign to stop the murder of the dead by an oppressive state, but feel free to correct me) life itself has an unpredictable albeit assured end that's (generally) out of your control - or anyone else's for that matter. A human right is an immanent principle (meaning it applies to anybody indifferently, while everybody's life is, hopefully, a very personal and varied experience) that holds its value even in time and places where it's not respected, and can't disappear, be lost or taken away completely. What's more, it doesn't pass away with you, unlike life.
I can see where you're coming from (I mean, apart from France) - you apparently wanted to take a cheap shot at the death penalty "in a democracy" (Hm, I wonder which one...) - but you've started with a wrong assumption.
"No state, no society should be allowed to kill its citizens.
That's why death penalty isn't compatible with a democracy."
Talk about a simplistic deduction... Anyway. So let's see: the "right" to live is not one (unless you're talking about abortion, but that's another debate). What is, on the other hand, is freedom and in the consecrated terms, "the pursuit of happiness", implying that we can seek happiness and wealth without the fear to be killed arbitrarily, either by the State or by another group of citizens or individuals among which we live.
In other words, it means we shall not be deprived arbitrarily from our life, because we own it. Only a slave doesn't own his life, as in fact he/she doesn't own him/herself at all.
See? There goes the right to ownership again, at the very root level of life, and the reason why it is the most important. It even precedes freedom actually, since self-ownership defines the grounds for liberty.
You have, as we just saw, the right not to be killed arbitrarily, including by the State, but - even before you reach that point - the right not to be searched, incriminated, arrested and prosecuted arbitrarily - all kind of things that do happen in totalitarian states.
In short, in any decent liberal democracy, and in the always possible event that you would violate other people's rights, you have the right to a speedy, public and fair trial by jury, with the assistance of counsel for your defense (Maybe that does ring a bell?) because, in the words of Thomas Jefferson, "Rightful liberty is unobstructed action according to our will within limits drawn around us by the equal rights of others.", one of the important terms here being "the right of others" something that your phony and self-centered "right to live" completely overlook implicitly.
As far as these conditions are respected (no arbitrary arrest, prosecution and conviction, and a fair trial) I can't see any reason at all why the death penalty wouldn't be "compatible" as you put it, with a democracy.
On the very contrary: since there is an ultimate crime (you acting according to your will without respecting the equal rights of others and killing somebody arbitrarily - whatever your reasons), then there has to be, naturally, an ultimate penalty (you being killed rightfully)
If not, the whole notions of "Rights" and "Law" (natural law of course) simply make no sense at all, as they can be violated at will.
"If the state can decide to kill you, then your "freedom" is a gift implicitely granted to you by the state."
Well, that's precisely what can and does happen anytime only in a totalitarian state (Saddam Hussein anybody?), where coincidentally "“ hold on, here goes a big surprise "“ you're not free indeed, and whatever freedom you think you have, actually is a gift from the said State, - and one that it can give and take back at will, of course.
That does not happen in any liberal democracy I can think about - and before you start to name some (Hm, I wonder which one...) try to make sure that "the state decide to kill" anybody... In doubt, check the part about "fair trial".
But then your point was democracy and the death penalty, not dictatorship and death penalty, right?
Unless you're really going to name that "democracy" you were thinking about, and it turns out to be one of the "popular" flavor "“ China, North Korea, etc.
Could it be one these you had in mind I wonder?