Root is currently England's vice-captain and is the overwhelming favourite to replace Alastair Cook, who resigned as Test skipper on Monday, February 6.
But fears have been expressed that captaincy will damage Root's batting, with former England off-spinner Graeme Swann insisting even the vice-captaincy has already been a burden as the 26-year-old is no longer "the cheeky, dynamic joie de vivre-type player he was two or three years ago."
Gillespie, himself a former Australia fast bowler, is convinced, however, that those fears are groundless after observing Root's development during five years as Yorkshire coach "I understand the point he's trying to make," Gillespie, who returned to Australia last year for family reasons, told the Guardian.
"And no one wants Joe to lose his swagger or that inherent cheekiness but this is a guy who is maturing all the time.
"He's played 53 Tests, he's 26 years old and he is ready to go," added Gillespie, whose successful time at Headingley included guiding Yorkshire to back-to-back County Championship titles.
"During my time in England, I picked up a slight reluctance to give young guys leadership responsibility. But looking at the set-up, I don't think there are any other candidates more qualified to take the role on than Joe Root."
While Root has only led Yorkshire in three Championship matches, Gillespie said a brief stint as acting skipper against Surrey last year convinced him of his tactical acumen.
"Our captain, Andrew Gale, went off for a few overs to see the physio, Joe took over and immediately brought himself on to bowl.
"That might seem like a bit of bravado but he simply weighed up the situation. There were two left-handers (Kumar Sangakkara and Steven Davies) at the crease, our main spinner Adil Rashid turns the ball into them, so let's try something different.
"He changed the field, brought catchers in, and lo and behold he picked up both wickets inside four overs.
"That was the kind of captaincy move that told me Joe won't be a guy who lets the game meander. He will be a leader who is always thinking two steps ahead."
Meanwhile, England great Geoffrey Boycott, renowned as a stubbornly defensive opening batsman, told the Daily Telegraph that "Root has to be more adventurous than Alastair Cook".
"Being careful is OK up to a point, but imagination, flair and taking a risk are absolute musts," added the Yorkshireman who was England captain briefly in 1978.
Root played for the same Sheffield Collegiate club as Michael Vaughan, who also had limited captaincy experience before becoming a successful England skipper.
"When Michael was appointed England captain, he had not been captain of Yorkshire.. yet he was brilliant because he was innovative," said Boycott.
If Root were to succeed Cook, he would be joining fellow contemporary star batsmen Virat Kohli (India), Steve Smith (Australia) and Kane Williamson (New Zealand) as a Test captain.
"It reminds me of the 1980s, where we had the four great all-rounders -- Ian Botham, Richard Hadlee, Kapil Dev and Imran Khan -- all competing on the world stage," said Gillespie.
"This decade we've got four great batsmen in Joe Root, Steve Smith, Virat Kohli and Kane Williamson and with them now all leading their countries, provided Root gets the job, to me it seems a similarly exciting time to be a cricket fan."