Recently, a humorous fan theory has suggested that the real reason Captain America couldn't fully lift Thor's hammer Mjolnir in Avengers: Age Of Ultron was because he rode his motorbike without a helmet, disqualifying him from true worthiness. Obviously, that's no more than intentional silliness, but it does raise the question again: why WAS Cap not deemed worthy by the hammer?
On the face of it, Cap is the most worthy figure in the MCU - far more than Thor, in fact - he is a vulnerable kid from Brooklyn who wants nothing more than to go to war to vanquish evil despite how easily he'd probably be killed, he's fiercely committed to his heroism and to the protection of his allies. He's respectful, morally perfect and honourable almost to a fault. He's basically the ideal human with a heart of gold wrapped up in a superhuman, hard-body casing.
So why wouldn't he be able to lift up Mjolnir, when every traditional marker for worthiness is met? Why would Mjolnir choose to find him unworthy?
Could it be a question of appetite for war? First, a little background on Odin, worthiness and Viking/Norse mythology...
3. How Odin Changed The Definition Of "Worthiness"
To understand Mjolnir's idea of worthiness, we have to consider the true meaning of the word. It shouldn't solely be considered in terms of Christian "good" or secular morality even, it should be considered in terms of what Asgardians (and the Vikings they were based on) would consider true markers of worthiness.
Norse mythology would value loyalty, bravery, a love of battle, aggression and a conqueror's spirit. In other words, everything that Asgard was built on and what Thor grew up believing was what was expected of him. Those were the same principals that Odin had grown with too and which he ultimately turned against for the greater good. It just took him a while to realise the impact on his son and how to solve the problem.
Up until he is stripped of Mjolnir and banished in his first movie, Thor lives up to those qualities. It's just that his father - who also previously did in his bloody conquest of the Nine Realms with Hela - changed what should be valued. He fundamentally flipped the Asgardian idea of worthiness from war-mongering and arrogant glory - everything Thor stood for - and made Mjolnir value other things like selflessness, the preservation of Asgard and the people of the Nine Realms and peace.
And that's why Thor ultimately proves himself worthy and why Captain America CAN'T.