There's no canon (TV or Film) mention of Spock's human family other than a fleeting statement in Discovery about a phrase that Amanda's mother used to use. It's impossible, therefore, to determine what Spock's relationship with his Earth family was although we do at least know that they were alive during his lifetime. It's logical to assume that he would have learned about them and possibly contacted them, but beyond that we really can't go any further.
In the Extended Universe of novels and stories there are a number of references to his family on Earth.
In Strange New Worlds: Family Matters Spock travels to Earth at the request of his aunt (Amanda's sister). His cousin was in a traffic accident and is comatose. We learn that his relationship with his human family is almost non-existent. He attended his grandmother's funeral and that's it.
“I—I almost didn’t recognize you; I haven’t seen you since you were a boy. Elizabeth wasn’t even born yet. How old were you then? About fifteen?”
He calculated as he turned up his collar; then he slid his hands back into his coat pockets. Wasn’t she going to ask him inside?
“Twenty-eight point seven two Terran standard years, Aunt Kathleen.” His breath formed a white cloud between them. She started as though suddenly aware of the below-freezing temperature.
No wonder she had transferred her hate to Spock. Physically, he bore little resemblance to his mother. Because he had already undergone the Kahs-wan coming-of-age ceremony by his first visit to Earth and Grandmother Grayson’s funeral, Kathleen had never seen the child-Spock laugh, cry, or play as a human child would. And now, he could not permit himself to demonstrate the emotionalism that would prove his mother’s heritage.
By the the end of the story he's on friendlier terms.
On another occasion (TOS: Ishmael) Spock travels back in time to the 1800s where he passes himself off as a human and spends time with his ancestors, presumably Amanda Grayson's great-great-(great?)grandfather and great-great-(great?)grandmother.