John Stamos has accomplished a lot in his life. But the one thing that he never expected for himself? Fatherhood.

"It was something I thought maybe wasn't going to happen for me," the actor, 58, told "Good Morning America." "I had the sort of attitude like, I've had so many blessings in my life and maybe this one isn't meant for me."

Stamos married his wife Caitlin McHugh, 35, in early 2018. Just a few months after the wedding, the pair welcomed their son, William "Billy" Christopher, who's named after Stamos' father.

"I really had to straighten up my life, and the universe said, 'OK, we'll give you a shot at fatherhood,' and then it happened," he said.

For first-time parents, or parents who may be attempting to have a child later in life, Stamos hopes to set an example that fear is normal, but he doesn't want it to stop anyone from becoming a parent.

"If men in their mid-50s are afraid of taking it on, I hope I'm setting an example," he said. "It's a little harder because he runs faster than I do, but other than that, it's the greatest thing in my life. I'm so happy I did it."

Stamos admitted that while the love for Billy, now 3 years old, was there instantly, it took "a while" for a connection to fully form -- and he wants other parents to know that's OK.

"Six months in or so I was like, 'I love my son, but I don't feel that connection that I always thought I would,' and [Bob] Saget said to me, 'Just wait until he laughs at you one time. You do a joke and he laughs at you,'" he recalled. "And then Billy did that, and we've been connected ever since."

One of Stamos' favorite parts of fatherhood is hearing his son laugh.

"Billy loves to laugh, and I love to hear him laugh," he said. "He's funnier than me already. He likes to do bits. … I love that he likes to laugh and that he likes to make other people laugh and wants other people happy."

To keep up with Billy and his own lifestyle, which includes touring with the Beach Boys, filming and recording his true crime podcast "The Grand Scheme: Snatching Sinatra," Stamos emphasized the importance of fueling up and eating healthy. It's why he's partnered with Arnold Small Slice, the breadmaker's new line of whole wheat bread that comes in smaller sizes.

"I run around too much, so I can't eat too many heavy foods," he said. "I like to have a little snack just to get me going. That's why the Small Slice is perfect -- it's perfect for Billy, and it's not wasting bread if you want to have a smaller version of what you have later in the day."

Throughout his lengthy career, Stamos has played a number of father figures. There's his iconic portrayal of Uncle Jesse in "Full House" and "Fuller House"; his stint as bachelor Jimmy Martino, who finds out he has a son and grandson, on "Grandfathered"; and his most recent role as Marvyn Korn, a father and basketball coach, on "Big Shot."

None of those, Stamos said, prepared him for what it would really be like to be a father.

"It's harder than I thought," he said. "It just weighs on you. Even when I'm sleeping, I'm thinking about him and his well-being."

The biggest tip Stamos said he can offer parents is to "overload on love, attention, patience and kindness" when it comes to discipline.

"Billy is 3 and 4 months, and he's checking his boundaries. He's pushing them as hard as he can," he said. "I think if we overload on all the good then he's going to learn not to do some of the things he does."

As an alternative form of punishment, Stamos and his wife utilize a "calm-down room" whenever Billy's emotions become too high.

"My wife is incredible," Stamos said. "She spends so much time learning, reading and talking to people to find the best way to raise our son."

"Being a parent is about putting your child's needs ahead of yours for the betterment of them," he added. "At the end of the day you don't get a trophy, you don't get a ring, you don't get a banner. But what you do get, if you do it right, is a more loving, kinder, smarter and better version of yourself."