A little bit of background info. My dad was born in 1945 and lived through all of this. However, he never fought in the war as he received some kind of waiver that allowed him to avoid the draft. He feels strongly that this war was just and that we definitely could have won.
We just began watching the Ken Burns Vietnam War documentary and in the discussion we were having after watching the first episode he hit me with the statement in the title. We were discussing how it was left leaning democrat presidents that first got us into the war and then I pointed out that it was the right leaning people who perpetuated and strongly supported the war even after it became clear that there could be no victory. He then angrily told me that there was no right wing at that point in our history. Is he correct?
Another bonus question that I would like to see answered: My dad claims that after the Tet Offensive was repulsed America was on the verge of winning the war but it was Walter Cronkite and the media that convinced the country that it was futile. Is that true? Having watched the Ken Burns documentary before, I know that it is portrayed as essentially a hopeless war right from the very beginning. My dad argues that Ken Burns is extremely biased on this subject, though.
Does your dad mean right wing movements organized en masse for the war, or right wing social movements generally?
There were plenty of right wing social movements during the Vietnam Era. The modern American Conservative movement really hit its stride during the Vietnam era. It's funny how much they actually understand themselves specifically as a social movement: to this day, it's common to hear them called "movement conservatives".
Though they started, arguably, in the immediate post-War period, when organizations and party-organs like the National Review start coming about, their success really does book end the Vietnam-era. The first movement Conservative presidential candidate was Barry Goldwater in 1964 (while the movement has become the mainstream in the Republican party, at the time even "mainstream" conservatives were highly critical of him as a "radical") immediately before the war and the first successful movement Conservative candidate was of course Ronald Reagan, in one of the first post-War elections. Throughout this period, the movement was growing. Lisa McGurr's Suburban Warriors: The Origins of the New American Right (focused on Reagan's own Southern California) is one of the best books on the subject as a social movement, with George Nash's 1976 the Conservative Intellectual Movement in America Since 1945 being probably the most important intellectual history of the Conservative movement.
Most contemporary social movement scholars think in terms of "political fields" where different actors position themselves vis-a-vis another. Movements don't exist in isolation from one another. The American labor movement helped marginalize American socialists after the War while mainstreaming industrial unionism, the Gay Liberation movement marginalized pedophilia while (eventually) mainstreaming the trans* movement (while drawing strength and inspiration from the Civil Rights movement), and the Conservative movement worked to marginalize more radical group like the John Birch Society, anti-Semites, and (eventually) McCarthites, while mainstreaming some reactionaries, like Phylis Schlafly. While Conservatives weren't rallying for the War, they were engaging in other forms of political activism, including things like rallying against the ERA.
Lastly, the most obvious right-wing (or small-c conservative or traditionalist or whatever language you want to use) movement of the Vietnam Era is the Klan and other segregationist and white supremacist organizations. Indeed, one of the best books on the Civil Rights Era Klan (focusing on North Carolina) is by a social movements scholar: Klansville, USA by David Cunningham. If you like, there was a good episode of the PBS show the American Experience based on the book. You can watch it on the PBS website or YouTube. His first book is about how the FBI treated the Klan and the New Left, it's called There's Something Happening Here, but it just goes to show how you can really treat the Klan as you would any other social movement.
As for pro-War movements, in the 20th- or 21st Centuries, I'm not aware of any real "pro-War" social movements. Certainly, the press tends to be in favor of wars, especially at the beginning, and mass media also similarly tends to treat their country's soldiers heroically (in the Vietnam era, think of the 1968 John Wayne movie the Green Berets, or earlier films like A Yank in Viet-Nam, or the 1966 #1 hit song the Ballad of the Green Berets--critics didn't care for these but they were certainly profitable and popular with audiences, and as the the Green Berets was in theater the mass media did become more skeptical of American prospects in Vietnam), but I can't really think of many pro-War movements. The tactics of modern social movements tend to advocate for policy or social change, rather than the status quo, and their repertoire of marches, vigils, rallies, pamphlets, etc. reflects that.
I see that /u/yodatsracist has answered your main question well; in regards to the reasons why America lost the way, we have an FAQ section on 'Explaining America's Failure' in Vietnam, which links to some previous answers including one by /u/bernardito, with a detailed answer to the question To what extent was counterinsurgency actually successful in Vietnam? - hope that helps!