Princess Anne, the only daughter of the late Queen Elizabeth II and younger sister of King Charles III, suffered minor injuries and a concussion during the incident on Sunday at Gatcombe Park estate.
The 73-year-old was taken to hospital in nearby Bristol.
"She is recovering well, thank you," her husband Sir Tim Laurence told a well-wisher during a visit to hospital on Tuesday.
"We are both profoundly grateful to the medical team and hospital support staff for their expert care - and to the emergency services who were all so wonderful at the scene.
"We are both deeply touched by all the kind messages we have received from so many people near and far. It means a great deal."
Buckingham Palace said Anne had been kept in hospital as a precautionary measure, but was expected to make a full recovery.
A royal source said the injuries she had sustained were consistent with impact from a horse's head or legs.
Anne was expected to return home this week, but upcoming engagements have been postponed and she will not fly to Canada as planned at the end of the week, the source said.
The Princess did not attend the state banquet in honour of the visit of Japanese Emperor Naruhito and his wife on Tuesday.
The emperor's trip, postponed from 2020 due to the Covid-19 pandemic, seeks to deepen the military, cultural and scientific links between the two nations.
The visit comes at a difficult time for the British monarchy, after Charles, 75, was diagnosed with cancer earlier this year, and his daughter-in-law Princess Catherine, 42, wife of Prince William, undergoes preventative chemotherapy treatment for cancer.
