Please don't use that. Aside from the atrocious grammar, we are Americans. We could refer to the British as "wankers" or whatever pejorative is in at the moment, but you call yourselves the British and so we respect that.
America is a content (two, actually), thus "American" describes someone from said continents, not necessary the most powerful nation on it. I'm well aware of the colloquial usage that "American" = "person from the United States of America", but keep in mind that parent poster was not trying to be disrespectful, but rather wanting to be technically correct.
Which, as Futurama would have us know, is the best type of correct.
There's no disrespect meant when the word "USian" is used, and as a Canadian it's a tiny bit annoying that you guys have unilaterally taken the term.
"USians" is technically more ambiguous since we're not the only "United States"--Mexico is "Estados Unidos Mexicanos", which is Spanish for "United Mexican States".
As you may or may not have noticed, the names of countries don't always exactly match every single geographical use of the same word. Some people in Greece were similarly upset about Macedonia, which is why lots of people call it the "former Yugoslav Republic of Macedonia". I don't think anyone ever calls their citizens "Former Yugoslav Republicans" rather than "Macedonians", but my Greek isn't up to finding out for sure.
If you want to go further back, "Germany" and "Spain" are Anglicizations of "Germania" and "Hispania", which in the original Latin incorporated much larger areas than these current countries. I don't think many Polish or Portuguese people really mind, though, when I don't refer to them as German or Spanish despite the fact that they live in Germania or Hispania.
 Historical reason: of the original 13 colonies, they each had their own names, and the only thing they really had in common were that they were United in opposition to the Crown, that they considered themselves States, and that they were of America. By the time they had become a single nationality, it was too late to pick a more interesting name.
If anything, my experience is that the Europeans I've interacted with use it a bit more widely than even Americans do. While we universally use it to describe the people, many Europeans seem to use it as a name for the country, "America", in places where I personally find it weird. Outside contexts like national holidays and poetry, I usually say "the United States" or "U.S.", as in "I'm flying back to the U.S. next week", but many Europeans I know say things like, "this is my first trip to America" or "I have a brother in America".
Personally, I use "American" in your intended way, mostly because it's been so damned long since you guys unilaterally took over the name that it hardly seems worthwhile fighting for it.
I'm frankly surprised that this it the first you've heard of it - I've seen the term "USian" used with no possible connotation of disrespect for years, often by Americans themselves. IMHO it's unbecoming that your first assumption upon seeing this word is to assume that he's hostile against your country.
The stereotype of the snobby European prick is about as tired as that of the ignorant American redneck.
Trust me on this one, if you've spent any time on the Internet you know it's a passive-aggressive ditch at the US (at least common enough to deserve a couple notes in Urban Dictionary). Anyway, when did it become the role of members outside a group to tell them whether or not they should be offended by the name they are called?
The problem is that the name "America" is hardly exclusive to the USA. It's as if a country called "The Grand Duchy of Europe" referred to its citizens as "Europeans", to the great dismay of everyone else in Europe.
This is completely irrelevant. Citizens of the USA have an official demonym in the English language and it is American. The argument that it is an attempt to avoid ambiguity is bullshit because no one ever uses the term Americans to refer to all inhabitants of the North and South Americas.
Um, when they started calling them names? Now, don't take this the wrong way, but that sounds a little defensive. It's quite normal for people to explain the reason they chose a particular characterization of a group, that name catching on, and the other group getting affronted at being called a group. At this point a new characterization, one without connotations, is found. All along, people keep explaining that there's nothing bad at being in a group, or alternately that grouping is only done to divide and conquer.
no one ever uses the term Americans to refer to all inhabitants of the North and South Americas.
And that'll be because they know they'd be misunderstood. I used to call people from the USA "people from the United States", but that got too cumbersome. So I switched to Yanks - fully aware that it is also a poor choice of word, but at least more specific and apolitical around here. I've most recently transitioned to "American" plus qualifiers (such as state), since I've been reading stuff written by themselves. I don't see any other words coming beyond the horizon, except maybe cultural ones.
However, I wouldn't put calling all inhabitants of the Americas "American" past biologists and other non-geopolitically minded people.
If the European Union ever turned into a real country rather than a loose confederation of allied states (just like the US did between 1776 and 1865), I bet you'd call citizens of that country "Europeans". Unless their respective countries had joined the EU in the meantime, the Norwegians, Swiss, Russians, Icelanders, Turks, Belorussians, Ukrainians, Moldovans, Serbs, Croats, Macedonians (or is that Former Yugoslav Republicans?), Kosovans, and Bosnians may or may not be annoyed at that usage.
 Some countries call Macedonia the "Former Yugoslav Republic of Macedonia", since some people in Greece are annoyed that they named their country after a region that extends into Greece.
To your point, American citizens have long emphasized the word America. Since the beginning, in fact: the original Declaration of Independence doesn't capitalize the term "united", calling itself the "unanimous Declaration of the thirteen united States of America" adopted by the "Representatives of the united States of America" on July 4, 1776.