The post comes hours after Saoirse Kennedy Hill was reportedly laid to rest following her death last week. Kennedy Hill, the granddaughter of the late Robert F. Kennedy, died after being found unresponsive last Thursday at the Kennedy Compound, officials said.
Schwarzenegger is the daughter of Maria Shriver, who is the niece of Robert, John and Ted Kennedy, while Kennedy Hill, 22, was the daughter of Paul Hill and Courtney Kennedy Hill, the fifth of 11 children of Robert and Ethel Kennedy.
"After wiping away the tears, after the beautiful sunsets, there is always a new day," Schwarzenegger wrote. "It’s not the big events or pretty things that hold you up at the end of the day, it’s family. I feel so incredibly blessed to be part of such a loving, supportive and strong family that shows up in good times and in tough times."
She added, "I love each and every one of them and am so grateful to have them in my life."
Friends and family like Cristina Cooks commented on the post, writing, "Sending love to you and your family! Keeping all of you in my prayers. So deeply sorry for your loss."
Molly Shriver wrote, "I love u Kathy."
At the time of her death last week, Maria Shriver wrote a lengthy post herself, with a graphic that read, "A brave young women left our world yesterday."
She said in a caption, "Life is fragile and heartbreaking. It turns upside down in a minute. Love your children, hold them tight. Love your family, hold them close. Love your friends, keep them near. Be gentle with others, as so many are fragile and struggling."
She then added, "Actually, I think it’s best to assume everyone is struggling, so treat everyone with love, tenderness, and compassion."
Kennedy Hill was public about her struggles with depression. In 2016, while still a high school senior at Deerfield Academy, she wrote an op-ed for the school's newspaper mentioning that she left school during her junior year for "treatment for my depression and returned to the valley for my senior year."
As expected, that post also earned unanimous support, with Jessica Seinfeld writing, "May her memory be a blessing. Holding you all in the light, Maria."
Vicky Cornell added, "So very sorry for your family’s loss. Love and prayers."
Shriver has been reflecting all week, adding on Twitter Sunday, "I’ve been thinking it’s time for us all to pause."
At the time of Kennedy Hill's death, the Kennedy family also released a statement.
"Our hearts are shattered by the loss of our beloved Saoirse. Her life was filled with hope, promise and love. She cared deeply about friends and family, especially her mother Courtney, her father Paul, her stepmother Stephanie, and her grandmother Ethel, who said, 'The world is a little less beautiful today,'" the Kennedy family said in the statement.
ABC News' Matt Zarrell contributed to this report.