"My family story isn't special," said Castro, 37, who made history as the first Hispanic to deliver a keynote address at the Democratic National Convention. "What's special is the America that makes our story possible."
The opportunity that America holds for its people is that "no matter who you are or where you come from, the path is always forward," Castro told delegates in the Time Warner Cable Arena in Charlotte, N.C.
Those who came to America's shores "believed the opportunity they created today would lead to prosperity tomorrow," Castro said. "That's the country they envisioned and that's the country they helped build."
He recognized the middle class as the engine of the nation's economic growth and said hard work should mean people can aspire to reach the middle class and beyond.
And the means to get ahead and prosper is education, Castro said.
"We know you can't be pro-business unless you're pro-education," Castro said of his administration in San Antonio. "We know that [pre-kindergarten] and student loans aren't charity, they're smart investments in the work force" of tomorrow.
Castro said he watched last week's Republican National Convention in Tampa, Fla., and listened to speakers discuss individual success.
"We all celebrate individual success," Castro said, "but the question is how do we multiply individual success? The answer is President Barack Obama."
Republican presidential nominee Mitt Romney "quite simply, doesn't get it," Castro said.
"We know that in our free-market economy, some will prosper more than others," Castro said. "What we don't accept is that some won't get a chance ... and Mitt Romney and [running mate] Paul Ryan are perfectly comfortable with that."
The Romney-Ryan budget "dismantles the middle class," Castro said.
And that's not all, Castro said. Whether it concerns getting the middle class back to work, women's reproductive rights or the opportunity to marry someone no matter the gender, "Mitt Romney says no."
And healthcare access?
"Actually, Mitt Romney said yes" while governor of Massachusetts, Castro said, "but now he says no."
People recognize "freedom isn't free," Castro said. "What Romney and Ryan don't understand is that neither is opportunity."
"In the end, the American dream is not a sprint or even a marathon, but a relay," the mayor said. "Our families don't always cross finish line in one generation, but passes on to the next the fruits of their labor."
"With the wisdom of our founders and values of our family, America prevailed. With each generation going further than the last, America prevailed. And with the opportunity we build today for a shared prosperity tomorrow, America will prevail," Castro said. "It begins with re-electing Barack Obama. It begins with you. It begins now."
Megyn Kelly: Santa Claus and Jesus are both white men
Susan Sarandon: I was stoned at almost every awards show