The presidents of Argentina, Ecuador, and Venezuela have said they will not recognize the government of
Francisco Federico Franco. Costa Rica deplored Fernando Lugo's removal (and offered safe harbor) but did not mention recognition. Bolivia, Chile, and Colombia expressed concern. UNASUR is sending a delegation. Across the ideological spectrum, there is a sense that the politically-motivated and rushed procedure was not a legitimate way to remove a president.
So will that matter? Unfortunately, the answer is likely no, unless governments are willing to strangle the Paraguayan economy or credibly threaten to do so. International relations is all about power, and strong rhetoric now can be massaged later. The outcry about Honduras in 2009 ultimately had no impact whatsoever. In fact, Dilma Rousseff's response now strongly echoes that case:
"What the ministers are trying to do is to create an environment that allows a less traumatic solution for democracy, since President Lugo has a mandate that expires in eight or nine months and cannot be re-elected," said Dilma Rousseff, president of regional powerhouse Brazil.
Translated cynically: an election is coming up, so let's figure out a way to stall until then.
Franco, meanwhile, said that "God and destiny wanted me to assume the presidency." He did not mention the boatload of corrupt lawmakers.
Colin Snider has a good discussion over whether to label it a coup. I agree with his assessment that it is an illegitimate way to remove a president, but it's not precisely a "coup."