Parent Kori Doty - a non-binary transgender person who identifies as neither male nor female - aims to allow the child to discover their own gender.
The health card has been issued with a "U" in the space for "sex", which could be for "undetermined" or "unassigned".
Kori Doty is fighting to omit the gender from the birth certificate.
The parent gave birth to Searyl Atli in November at a friend's home in British Columbia. Kori Doty, who prefers to use the pronoun they, argues that a visual inspection at birth is unable to determine what gender that person will have or identify with later in life.
They want to keep Searyl's sex off all official records.
"I'm raising Searyl in in such a way that until they have the sense of self and command of vocabulary to tell me who they are, I'm recognising them as a baby and trying to give them all the love and support to be the most whole person that they can be outside of the restrictions that come with the boy box and the girl box," the parent was quoted by CBC as saying.
Kori Doty, a community educator who is part of the Gender-Free ID Coalition, said that those who feel different to the gender designation assigned at birth face several problems later in life trying to change their documents.
"When I was born, doctors looked at my genitals and made assumptions about who I would be, and those assignments followed me and followed my identification throughout my life," Kori Doty said.
"Those assumptions were incorrect, and I ended up having to do a lot of adjustments since then."
In the case of Searyl Atli, the parent said that the authorities have refused to issue the birth certificate without a gender designation and so the parent has applied for a judicial review of the case.
The family's lawyer, barbara findlay, who chooses to spell her name without capital letters, told Global News: "The assignment of sex in this culture is done when a medical person lifts up the legs and looks at the baby's genitals. But we know that the baby's own gender identity will not develop for some years until after they're born."
Canadian media reported that the baby's health card could be the first in the world to not have a gender designation.

The Japanese engineer created many popular drum machines, including the iconic TR-808.
Its sound is a staple of hip-hop and electronic music, used by everyone from Kanye West to Marvin Gaye.
Kakehashi received a technical Grammy in 2013 for contributions to electronic music technology.
Dave Smith - Kakehashi's co-winner - told the BBC he "was just an amazing man, a good friend, a very good competitor of course and just innovative continually all that time".
Before leading Roland for 40 years, Mr Kakehashi founded Ace Tone in the 1960s.
The firm made amplifiers and primitive drum machines, laying the groundwork for the engineer's future success.
The sound of the TR-808 proved a game-changer in the 1980s and 90s.
It appears on Marvin Gaye's "Sexual Healing", and in the opening bars of Whitney Houston's "I Wanna Dance with Somebody".
Rapper Kanye West's 2008 album 808s & Heartbreak showcases the machine throughout.
Musicians have shared their tributes to Mr Kakehashi online, calling his impact immeasurable.
Marc Almond of synthpop duo Soft Cell called him "a man who changed music".
Martyn Ware, who played keyboards for The Human League, told the BBC: "Roland was central to everything that we did for the whole of the first two albums - they featured on every track."
"We were loyal to them like people would be loyal to a football team."
In 2015 the TR-808 was the subject of a documentary featuring Pharrell Williams, Fatboy Slim, and Phil Collins - among many others.
In a farewell to Mr Kakehashi, his colleague for almost four decades, Roland developer Tommy Snyder wrote: "He was a super funny, wonderful and gifted human being, and his contributions to the musical instrument world, and music, touched millions of people worldwide. RIP dear Taro."

Three houses were searched in Lisburn, Belfast and Antrim in connection with the seizure.
A man interviewed by police was later released, pending further enquiries.
Detectives worked alongside officers from the National Crime Agency in the operation.
An eight-month old Canadian baby has been issued a health card without a gender marker, in what could be the first case in the world.
[NEXT_CONCEPT]
The music world is mourning the loss of Roland founder and electronic instrument pioneer Ikutaro Kakehashi, who has died aged 87.
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Police say they have seized herbal cannabis worth £100,000 that was destined for Northern Ireland.