When it comes to the comic books, the answer to this question seems to hinge on a few more things than it does in the Marvel Cinematic Universe—at least so far. In the latter, the stated reason is that only those who are worthy to lift the hammer will be able to do it.
However, worthiness, in the comic books, anyway, is up to interpretation, especially in the What If Marvel books, which seem to be able to bend the rules at will. Those books aside, there are probably other reasons why others, and Captain America in particular, are able to wield Thor's weapon.
In Avengers: Age of Ultron, Steve Rogers actually moves Mjolnir ever so slightly during an after party contest that sees every Avenger trying to lift it except for Black Widow, who says she doesn't need to know if she can or can't. Then, Ultron shows up and ruins the mood.
However, later in the movie, after Vision is created, has a brief skirmish with the heroes, gathers itself and apologizes, the Avengers aren't all that sure if they can trust a new life form that is at least part Ultron (and also part Tony Stark, another character falling out of trust).
To settle the matter, though, Vison suddenly picks up Mjolnir, while saying there's not much it can say or do to convince them it's on their side, and hands it to Thor. That opens up a debate as to whether or not an artificial intelligence/machine can actually be worthy (since if left on the floor of an elevator, presumably, the hammer would move as the elevator went up or down).
With that in mind, here is why I think comic book Captain America has lifted the hammer (as well as some others, and this will also apply in some cases to Vision in the movie), while MCU Cap did not.
Worthiness isn't the only reason. That appears to be a starting point, but in the case of Captain America, with one story, Thor was down and Cap used the Mjolnir to bash away some bad guys before throwing the hammer back to Thor. In another, he used it to rally the Avengers against a particularly fearsome foe.
In Age of Ultron, Rogers may well have been worthy (which is why he even budged it at all), but it wasn't needful for him to do anything more.
In the case of Vision, it wanted a rather quick and effective way to end the debate over its intentions, which apparently Mjolnir, or the spell guarding Mjolnir, was able to sense and thus provide a way for the detractors to put their differences aside for the moment and go after Ultron.
in the comics, it seems more of a contrivance than anything else. You've done so many stories, what else is there to do? It tends to lend itself to shock value when someone other than Thor is able to wield the hammer. It provides a pretty cool moment, one that hopefully is worth more than the moment itself, but can provide some kind of reverberation throughout the Marvel Universe. I'm not sure if that's been the case or not, other than fans thinking it's cool that Cap did it here, or someone else did it there.
This seems to be the way most of the aforementioned What If issues went. I don't think we can consider any of them as mainstream cannon, however, and in reality, unless the story takes place somewhere in the multiverse, they probably shouldn't even count, since not even within the context of reality was it possible, but instead, a pure what if scenario.
I'm just going to throw that out there, for the cynics, and because, let's face it, it's probably true. It happens once, it could certainly happen again, but over and over and over gets a little old fast. At the very least, Mjolnir's special weapon status suffers. Besides, does the hammer have to be the answer to everything? Is there no other way for people to get out of the predicament they were in? It could be (and this is pure speculation and should be taken with the smallest grains of salt), that the writers wrote themselves into a corner and giving Mjolnir to Captain America for at least a few moments was considered an acceptable way out. Because, well, he's Captain America.
WILL IT HAPPEN IN THE MCU?
Since we now no Mjolnir cannot only be held by someone who you would think would not be considered worthy, as well as be utterly destroyed, thanks to Thor: Ragnarok, I guess we won't get to see if Captain America can wield Mjolnir. But who knows, maybe he will show his worthiness, and the need, by picking up Stormbreaker. And maybe Cap will have the presence of mind to aim for the head, too.
I’m pretty sure you could see the hammer lift ever so slightly in the movie which suggested that, with a little more effort, that the Cap would be able to lift the hammer.
It seemed to suggest that in a future movie we would see the Cap actually lift and use the hammer which is what we saw in the comics.
That’s my prediction anyway. And given the significance of the moment in the comics, it would be pretty lame if the moment didn’t make it into the franchise.
The hammer is not just any hammer. Maybe it's the right opening sentence to reveal the secret.
Then, how heavy is Thor's hammer?
According to the story in the Marvel comic edition Thor's Hammer (1991), it is said that Uru's own fictional metal weighs only 42.3 pounds or about 19,187 kilograms. So, if we calculate the volume weight, it is estimated to be around 2.13 grams / cubic centimeter.
Then why, in the Avengers: Age of Ultron, Captain America is so overwhelmed to lift Thor's hammer?
So in this case, the scientific reason behind the inability of Captain America is to lift Thor's hammer because a stationary object will remain stationary if there is no electricity network acting on it. The key to explaining the inability of the American Captain to lift Mjolnir is the word in the last sentence, namely 'network'.
When the hammer is placed on the table, there is a downward pull on it, namely the gravity between the hammer mass and the earth's mass, which is called the gravity weight and the opposite force of the table that supports Palu. The opposite force in physics is called the normal force which acts in a perpendicular line from any surface. Basically, this style is natural electrostatic force.
So, when using the hand, it will produce a repulsive force upwards, greater than the weight of the hammer, but the hammer stays still. Then where does the extra gravitational power come from? He concluded that origin was a metal element of Uru originating from the star's core or planet. So, if it really comes from there, then the hammer automatically has the power of graviton aka extraordinary graviton - in superhero science terms.
Logically, when an object has an additional gravitational force, it can increase the weight of the object. Based on that reason, when they tried to give the force of repulsion upward, the Uru metal immediately increased the weight of the hammer so that it could not be lifted. Hammerheads remain motionless. Heavy loads will not damage the table, because the Uru metal will only remove enough gravitons to balance the lift so the hammer stays silent. After the lifting force stops, the emission of hammer gravitons at that moment also disappears.
Uru's metal ability to change interactions with gravitational fields also adds to Mjolnir's other specialties, that is, when thrown it can change direction, target and move, like his own volition, then return to Thor's hand. Such is the explanation in science.
He wasn't worthy at that Time.