The director, behind hit documentaries like 'Fahrenheit 9/11', 'Bowling for Columbine' and 'Roger & Me' among others, is known for his anti-gun views.
Eastwood's film, starring Bradley Cooper, is based on the real-life story of Navy-Seal Chris Kyle, a deadly sniper deployed to Iraq.
"My uncle killed by sniper in WW2 (World War II). We were taught snipers were cowards. Will shoot u in the back. Snipers aren't heroes. And invaders r worse (sic)," Moore tweeted without naming the movie.
The Palme d'Or-winning director praised 'Selma' in his earlier tweets. The film's lead actor David Oyelowo missed out on a best actor Oscar nomination while 'American Sniper' star Bradley Cooper earned two nods -- one as the film's producer and second in the lead actor category.
"Everybody else - try to see SELMA this weekend! Simply because it is a piece of brilliant filmmaking, the likes of which are so rare...," Moore tweeted about the film on Martin Luther King Jr.