Specifically, what condemned Maximilian was sovereignty as a policy, belief and revolutionary stance in general, and not simply Juarez or the people of Mexico. Any territory unwilling to recognize the right of Mexicans to sovereignty over Maximilian's monarchy would be deprived of diplomatic relations with Mexico.
Radical Republicans in the U.S. Congress would take up the same arguments. In his address to the House on July 4, 1867, two weeks after the execution, Republican Congressman Zachariah Chandler argued that Maximilian was not only a "filibuster" beyond the bounds of legality, but that his "election" in Mexico had been as ridiculous as his governmental control over the country. And, most importantly, Maximilian's 1865 decree condemning to death anyone who took up arms against the empire, which threatened imprisonment within 24 hours of the decree's promulgation, equated his occupation to a dictatorship that went against all laws of family, society and humanity. The decree, "barbarous" and "inhuman," allowed that "the mother protecting her own child, engaged in patriotically fighting for her government and country, could be executed by the arresting officer. Had there ever been such a decree in a civilized age?"
[...]
In the same session of Congress, Republican Congressman James Nye went even further and pointed out the hypocrisy of Europe and the U.S. in expressing any opinion about Mexico's treatment of its wartime enemies:
"[...] what business is it of ours or of the governments of Europe how Mexico treats its adversaries? We would not allow them to interfere in our affairs. Suppose, for instance, that come the final encounter in our country, when Grant had them against the wall at Richmond, England, France, Prussia and Austria had got together and said, "Blood is going to be shed here; some of you are going to die, stop where you are, wait for the sake of humanity," or that they said, as in fact they did say, "You can get Mr. Davis, and if you do, for God's sake, don't execute him." What would our people say to that? "Stand aside; we will solve our own problems: we are the judges of our own affairs."
[...]
Even Nye, who had ostensibly defended Mexico's right to "mind its own business," concluded his speech to Congress by emphasizing the fact that Mexico's autonomy depended, ironically, on U.S. control:
"[...] on the map of this continent it is written that [Mexico] is ours, and we are going to have it [...] do not permit a foreign caretaker for Mexico; if it needs one, we will take care of it ourselves. We will not try to establish an imperial power, but we will try what we can do: to raise the average of their intelligence and increase their love for republican institutions. For now, the obligation of the United States is that of a great teacher; in fact, I may say that today the United States is a great missionary [...] that, sir, is the way I would conquer Mexico. I would conquer it with our divine principles.