It appears it is a GoDaddy-specific problem:
To solve the problem, I tried to access the .htaccess file using the "daemon" user, which is the one that launch apache2.
$ sudo -u daemon ls -dl /opt/bitnami/apps/gitlab/htdocs/public/uploads
drwx------ 4 git git 4096 Dec 9 06:58 uploads
As "daemon" only needs to go into the directory without listing the directory, it is solved by:
$ sudo -u git chmod 711 /opt/bitnami/apps/gitlab/htdocs/public/uploads
Also, the .htaccess didn't exist. I created a simple one (with rights 644) like the one from stackoverflow:
Allow from all
Require all granted
However, this makes the upload directory available to anyone who knows an URL inside it.
Even a non-logged user could access it.
Is there a way to explain in the .htaccess that the directory is only accessible to gitlab logged users ?