David Clark, who was in his 50s, was with his pets in a field near the town when the accident happened just before 7pm on Monday night. Air ambulances attended but he was pronounced dead at the scene.
Mr Clark, who was married with children, was a keen cyclist and rugby coach who had worked at Richmond School for 23 years.
His death was confirmed in an email sent to parents and staff by headteacher Jenna Potter.
A statement from the school said: "Dave Clark was the 'heart and soul' of Richmond School. He was an enormous character, a brilliant school leader and simply a lovely man who enriched the life of everyone he came into contact with, just by being himself and doing what he did every day.
"Dave was appointed to Richmond School in 1997 as Head of Key Stages 3 and 4 and teacher of PE. He was highly regarded by everyone he taught and families appreciated his firm but fair manner and the very high standards he expected of every student in the school.
"He was promoted to the role of Deputy Headteacher in 2003 and revelled in the challenges the post presented. Dave was always the person we looked to to deal with difficult issues and he was a master at that. He never let students get away with being anything other than their very best, never giving up on them, even when at times it felt as though they wanted to give up on themselves. All of this, Dave did with a smile and a level of kindness and care that is seldom seen.
"Outside school, Dave was a keen sportsman. He had grown to love the Dales and would often be seen cycling, walking and spending time on the moor during the beating season.
"Our students and their parents have been lucky to have known and been helped by him. We have been privileged as a staff to have worked with him and will continue working in the same way in his memory.
"Above all, Dave was a family man. Our thoughts are with Dave's wife, his children and wider family at this difficult time."
North Yorkshire Police said the Health and Safety Executive had been informed.
Mr Clark was a member of Richmond Cycling Club, who posted a tribute to him on their Facebook page: "It is with immense sadness that we learned today that one of the club's most respected and longstanding members, David Clark passed away due to a tragic accident. David was walking with his dogs on the outskirts of the town, off the town’s Hurgill Road, on Monday evening when the incident happened. Our thoughts are with David's family and friends who must be devastated and within the club there is deep sadness and shock."
He is the second person to die in a cow trampling incident in North Yorkshire this summer.
An 82-year-old man from Lancashire was killed near Ribblehead Viaduct in May as he and his wife walked their dogs. She escaped with minor injuries.
On July 19, a couple were injured by cows while walking through a field near Huggate in the Yorkshire Wolds. One had to be airlifted to hospital.
In the same month a couple in their 50s were also attacked in Upper Wharfedale, but escaped serious injury.