Kasich's statement about not supporting Trump because of a "matter of conscience" shows that Kasich is thinking intelligently about the direction the Republican Party is taking. It's not a matter of him being a sore loser, it is a matter of Kasich taking the high road and refusing to let Trump's inflammatory rhetoric define what it means to be a Republican.
John Kasich is not the type of politician who would be a "sore loser." He was one of the most respectable and rational Republicans in the primary race. He is not supporting Donald Trump because he recognizes that Trump is an awful presidential candidate. Trump is immature, egocentric and just generally not presidential material.
John Kasich has stated he is not supporting Trump due to "a matter of conscience, and I believe he is justified. Donald Trump doesn't care about anyone but himself. He makes personal attacks on people just because they don't agree with him on policy. He has no experience in politics. Ironically if the Republicans has nominated Kasich rather than Trump they would have a better chance of winning. Trump has many more enemies than Kasich, and it would only hurt Kasich, if he supported Trump.
John Kasich does not support the same political beliefs and values as Trump. He is not simply a rival who does not want his opponent to win. Even though it may seem like he has many of the same values, there are certain ones he does not share and will not endorse.