It also has one of the country's most advanced prosthetic workshops, dealing with 50 patients who have lost limbs since opening in June 2006.
Mr Browne said Headley Court was a "very special place" and that its facilities and care were admired around the world.
He said: "I am always overwhelmed and humbled by the bravery and courage of the troops I meet here, some of whom are overcoming complex and devastating injuries.
"This place is a combination of charitable trust and government support. The excellence of this place is as a result of that combination.
"Already minor miracles are being worked here every day and we are able to return troops back to operations. We need to maintain and build on that.
"But that doesn't mean we can't improve it and that is why we are investing in it today."
Among those being treated at the facility is Anthony Makin, 21, from Benwell, Newcastle, who is due to return to front line service after losing his leg in a roadside bombing in Afghanistan.
The commando is believed to be just the second amputee to return to active duty.
After having his right leg amputated below the knee, being fitted with a prosthetic limb and sent for rehabilitation at Headley Court, he has returned to 3 Commando Brigade of 29 Commando Royal Artillery with a promotion to Lance Bombardier.
Princes William and Harry visited the centre last month
He expects to serve with his regiment on their next tour of duty in October.
Mr Browne said it was such cases that proved the "important role" military rehabilitative health services played in the lives of service personnel.
The £24m of additional investment would ensure the reputation of the centre was retained so that it could treat others like Lance Bombardier Makin, he said.
"He is a fine example of the bravery and selfless commitment the British armed forces demonstrate daily.
"His personal battle to overcome his injuries is outstanding and it is clear that the treatment he received from the staff at Headley Court has been first-class, enabling him to carry on with the job he loves, and does so well."
But Diane Dernie, whose son Lance Bombardier Ben Parkinson was severely wounded in Afghanistan, told the BBC that the decision to award funding now was an example of Britain's "woeful state of unpreparedness" for the consequences of going to war.
"While Headley Court is punching way above its weight - it's achieving miracles - this award is now the government playing catch-up and trying to provide the facilities that should have been there before we embarked on this course of action," she said.
The charity Help for Heroes led the campaign to increase funding for Headley Court.
Founder Bryn Parry told the BBC: "It's an amazing place. At the moment it's a bit run-down because of the number of people going through it, but with this new money I think they'll bring it up to a wonderful standard.
"The whole ethos - the sense of humour, the feeling of push, the guys getting back - is just wonderful, and when you go there you can't help but be quite humbled by what they are going through."
The centre at Headley Court began its work after World War II for RAF personnel, but in 1996 became the main UK military rehabilitation centre for all three armed forces.
The site provides consultants, physiotherapists, occupational therapists, speech and language therapists, social workers, a psychologist and a cognitive therapist.
Princes William and Harry met patients at the facility on a visit last month.
During the tour, Harry was reunited with one of the injured servicemen who shared his return flight from his tour of duty in Afghanistan.
Marine Ben McBean, 21, lost an arm and a leg when a mine exploded during a patrol in the country, and he was flown back to the UK for medical treatment at the same time as the prince left the country.
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