Keep in mind that this is just a bill that hasn't passed into law yet ("avant-projet de loi"). And yeah, there's an exemption in there for medical conditions, even though it's not been discussed in the press.
un justificatif de statut vaccinal concernant la covid-19 [...]
Les personnes qui justifient, par la présentation d'un certificat médical, d'une contre-indication à la vaccination sont exemptées des obligations d'immunisation mentionnées au I.
a certificate of vaccination status regarding covid-19 [...]
Individuals presenting a medical certificate indicating a contraindication to vaccination are exempted from immunization obligations mentioned in I.
As the details of all this stuff, e.g. how the checks are going to be performed by e.g. supermarkets are still being ironed out, don't expect more detail than that on what the exemptions are going to be or how they are going to recorded.
As @Jontia mentioned in a comment, after a closer look, the exemption via a medical certificate ("doctor's note/excuse" in the more familiar English term[s]) seems to apply only to a section that deals with mandatory vaccinations for professionals who interact with the public. (The wording is in fact similar to that that deals with other vaccinations in the present 2013 law on that.) As the law proposal stands now, it seems there would be no exemption e.g. from presenting the alternative negative Covid-19 test result in other circumstances, e.g. going to an event or whatever else the French legislators will decide this health pass is going to be used for, but only for [continued] employment purposes.
(Among other lacunae of the current provisions, even EU-approved vaccines administered abroad don't count towards this French pass...)
In more recent developments, there's now an "avis" of HAS (the "High" public health authority) on this law proposal. It outlines that indeed the law would provide an exemption:
Le projet de loi prévoit :
des exemptions en cas de contre-indication à la vaccination. Ainsi, « les personnes qui justifient, par la présentation d'un certificat médical, d'une contre-indication à la vaccination sont exemptées des obligations d'immunisation »;
but they don't seem to discuss that the exemption seems to be only for employment purposes... although they do spend like 3 more paras recommending that the mandate for obligatory vaccincations should be enlarged. I guess they consider that the going-to-events bit is not mandatory... or something like that.
I'll also note from related articles that the government has had it some of its Covid-19 laws overturned by the Constitutional Council, e.g. the one dealing with religious events (this might sound familiar to the similar lawsuits in the US).
Since the start of the pandemic, the government’s Council of State has consistently underscored the importance of freedom of worship, which is protected under France’s constitution.
In November 2020, France's highest administrative court overturned the 30-person maximum capacity for religious ceremonies, a restriction initially imposed by the government. In their decision, the judges considered that the activities carried out in places of worship "are not of the same nature" as those in cinemas or theatres and that "the fundamental freedoms at stake are not the same".
So it's still somewhat unclear how much of this will hold water. As also noted in the French press, the Constituional Concil did give the go-ahead in May to the current scheme of the health pass... precisely because it only applies to "grand gatherings". (The left-wing deputies had submitted an objection that the law was too broad.) The CC also held precise enough the notion of "entertainment event" to exclude e.g. political gatherings; see #18 in full text. So it's a bit unclear if the Constitutional Council will approve of the health pass requirement for the "lesser" everyday things that the new proposal wants to apply this to (restaurants etc.)