June 18, 2018 05:20 PM

John Stamos has played a longtime bachelor-turned-grandfather on Grandfathered and the beloved Uncle Jesse on Full House, but his most fulfilling role yet is being a dad to 2-month-old son Billy.

“My first Father’s Day means that I don’t have to hand a kid back to the parents when someone yells cut. It means when people ask me if I’m a father, I don’t have to use the goofy line, ‘No, but I play one on TV,’ ” Stamos said on Instagram Sunday along with an adorable father-son portrait.

“My first Father’s Day means that from this day forward— I will start to look my age (and older). Most importantly, it means that for the rest of my life, I’ll always be known as someone’s dad. And that’s all I ever wanted. Happy Father’s day,” he added.

Stamos became a father for the first time on April 10 when he and wife Caitlin McHugh welcomed their son William “Billy” Christopher into the world.

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“Happy 1st Father’s Day to the best new daddy! I knew you’d be a great father, but you’ve exceeded expectations. You change diapers! Lol, and so much more, of course. We love you most,” McHugh said in a tribute to her husband on Instagram Sunday.

RELATED VIDEO: New Dad John Stamos Opens Up About Fatherhood & Reveals Which Parent His Son Looks Like

And after a celebratory Father’s Day weekend, Stamos is looking forward to spending more bonding time with his son.

The actor will be bringing Billy along on his first cross-country trip as the Fuller House star/producer returns to host PBS’s annual broadcast of A Capitol Fourth, the national 4th of July celebration in Washington, D.C.

RELATED: John Stamos, Joseph Duggar & More Celeb Guys Who Became Dads This Year

John Stamos

It’ll be baby Stamos’ first cross-country excursion, coming on the heels of a slightly not-so-smooth local road trip.

“It was Mother’s Day and Caitlin’s birthday, so we went down to Laguna [Beach], and it was the first time we didn’t have a nanny — that was rough!” he told PEOPLE in late May. “[But] so far, it’s been a dream, and when I look at him, I feel like I’ve been thinking about that face for a long time.”

RELATED: John Stamos Admits He Never Thought He’d Have a Child: ‘I’ve Cried Most Days’ Since Son’s Birth

Stamos also said that fatherhood has been everything that he expected it would be, and it’s deepened his connection to his country.

“I always knew it was going to be like this,” he explained. “It took me longer than it should’ve, but I knew that I would be a family and be contributing to this country, and holding onto the morals and the values that my parents had, that I like to think of the world having. It’s a time to be kind and graceful to people. It just takes so much energy to block all the negativity. I’m really trying to get people together.”

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