The 24-year-old said the gift, which will pay for three pieces of medical equipment, was her way "giving back to Barbados".
"I believe that this will have a huge impact on the people of Barbados. This was all done to save lives or at least extend them."
The singer's grandmother, Clara "Dolly" Braithwaite, died in June.
She had fought a long battle with cancer.
The hospital's radiotherapy unit has been renamed the Clara Braithwaite Center for Oncology and Nuclear Medicine.
Rihanna, whose real name is Robyn Fenty, made the announcement at a visit to the hospital over the weekend, where she was accompanied by her mother, Monica, and grandfather, Lionel Braithwaite.
She was particularly close to her grandmother, and took a last-minute break from her European festival tour in June to spend some time with her in hospital.
When she died on 30 June, the pop star took to Twitter and posted several photographs of the two together, writing, "my heart smiled and cried all at once".
Born and raised in Barbados, Rihanna is one of the most successful female musicians in the world.
Since her debut in 2005, she has released an album a year, with the exception of 2008.
Her most recent, Unapologetic, became her first US number one, and gave her a global hit with the lead single, Diamonds.
Forbes magazine last week named her the third-highest-earning woman in the music industry, after she made $53m in 2012.
Pop star Rihanna has donated $1.75m (Â£1.08m) to a hospital in Barbados in memory of her grandmother.