Its not an anti-american signal (it wouldn't be taken any more seriously than if timbucktoo's dictator waved his arms shouting "death to America!").
Its intended to show a variety of folks that France has nukes and that despite underfunding (nay, starving)their military capabilities for 60 yrs now, Europe still has some muscle that counts etc.
First, Chirac was defending the nuclear deterrent from talk of budget cuts. Second, he wasn't warning America of anything but I'm sure they gave this explanation to appeal to anti-American Europeans. That crowd could never understand just threatening terrorists who would kill inoccent French citizens, but if it's to one up America, right on.
Muscle without will means nothing, old Europe is DEAD.