Jipur is a strictly 2 player game, plays very fast. It has hand management, so it is a little competitive but I have found it to be relaxing.
Carcassone isn't a 2 player, but it plays very well with 2 players. Very relaxing game.
Patchwork is a must for 2-player non competitive games. The theme of making a quilt looks boring, but it is quite fun.
Battle Line and Lost Cities are two games by the same designer, both are 2 player games that are a bit stressful. Not directly competitive in the sense of a war game but you are trying to beat the other person. They are both brilliant games though.