A National League South match between Bath and Dulwich Hamlet at Twerton Park was abandoned after five minutes when striker Fletcher collided with a hoarding.
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He was soon taken to hospital by ambulance, and underwent emergency surgery following the worrying scenes.
On Wednesday morning, Fletcher's parents released the following statement via the club.
It read: "Unfortunately Alex has sustained a severe head injury as a result of last night's events.
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"He required emergency neurosurgery to stabilise his condition and was then admitted to intensive care where he remains in a stable but critical condition.
"Alex's family would like to thank everyone for their best wishes that have been sent."
The club added: "No further comments will be made at this stage and we ask you all to respect the family's privacy.
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"Everyone at Bath City is sending their unconditional love, strength and support to Alex and his loved ones."
The scoreline was 0-0 when the incident happened and after 25 minutes it was decided the match would be abandoned due to the serious nature of the injury.
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