The most important thing that makes someone a Romantic hero is his attitude towards society and its conventions. Romantic heroes are rejects -- either society has rejected them or they have rejected society. These heroes are extremely unconventional.
This characteristic can easily be seen in the character of Paul Baumer. Baumer's whole character is formed by his attitude towards the war in which he is fighting. Baumer rejects all conventional attitudes about the war. He does not feel heroic. He does not glory in killing the enemy. He hates the war and does not feel that it has any greater purpose.
By holding these attitudes, Baumer distances himself from the conventions of his society. This makes him a Romantic hero.