When Goku first fought Beerus during the first arc of Dragon Ball Super, it was clear that, while Super Saiyan God Goku could stand up to Beerus, the God of Destruction vastly outclassed him. However, since that point, Goku has only increased his power. He first learned how to harness the power of the Super Saiyan God Super Saiyan -- more commonly known as Super Saiyan Blue. Then, he unlocked the full extent of Ultra-Instinct during the events of the Universal Survival Saga.
By the end of the Galactic Prisoner Arc of the Dragon Ball Super manga, Goku has seemingly mastered the Ultra-Instinct Form, achieving such mastery over it that he even pushes Whis with his sheer power. This all begs the question: can Goku beat Beerus now, after the Moro Arc?
How Powerful is Goku With Ultra Instinct?
Dragon Ball Super established that, even with Super Saiyan Blue, Goku could not defeat Beerus. However, Goku has continued pushing the limits of his strength, increasing his potential of beating the powerful God. When facing the strongest champions selected by the various Gods of Destruction, Goku had to use his Ultra Instinct abilities against Jiren, one of the mightiest figures in all reality.
Ultra Instinct is what makes Goku the real threat to Beerus, as it is a technique that gives many of the Gods and Angels of Dragon Ball's universe an edge -- allowing them to act without thought. It puts Goku on par with Beerus to an incredible degree. Even though Whis easily beat a fully-powered Goku in Ultra Instinct mode after the battle with Moro, Whis is staggered a little by Goku's newfound power. Whis is superior to Beerus, which means that Goku proved a match -- if only briefly -- against someone even stronger than Beerus.
On top of that, Beerus isn't necessarily the strongest of the Gods of Destruction either. He was able to compete with five of the other Gods of Destruction during the Universal Survival Arc -- potentially coming close to killing them -- but that doesn't mean he's the strongest even in his own group. That means that Goku can, in theory, achieve Beerus's level of power with enough training and potential.
Beerus's Unseen Power Would Destroy Goku
That said, Beerus has only used a fraction of his power against Goku and Vegeta. When Vegeta fought Beerus later on as a fully-powered Super Saiyan Blue, the moment Vegeta hurt Beerus, he thrashed him in microseconds in retaliation. Over the course of Dragon Ball Super, Goku and Vegeta have never fought Beerus at full power.
While some have speculated that Super Saiyan Blue Vegito or Gogeta might surpass Beerus in terms of power, it's just that: a suspicion, not a confirmation. Many rumors surrounding Beerus end up failing to capture the whole picture. Goku states that Broly at his peak might be stronger than Beerus, but, again, it's just Goku's speculation. Much about Beerus's strength is the subject of rumor, in part because we rarely experience his raw power. The most memorable demonstration of Beerus's power showcased him using perhaps 70% of his strength, and this proved to be enough to almost break the universe.
With the raw force of Ultra-Instinct, it is possible that Goku can compete with Beerus. However, while this may push Beerus to use more of his power until we can see the limitations of the God of Destruction's strength, it is impossible to really judge just how powerful he is. Plus, if Broly's strength is comparable to Beerus's, that makes it even less likely that Goku is on Beerus's level -- especially since Goku needed to fuse and become Super Saiyan Blue Gogeta to even compete with Broly.
In essence, until Beerus fights Goku at 100% of his capabilities, it's impossible to determine if Goku can win. Still, Goku is closer to beating Beerus now than ever before. While the gap in their abilities is undefined, all signs point to Goku needing a lot more training before he can take down Beerus.