TL;DR: No. The rule concerning tax domicile, in itself, neither authorizes nor prohibits you from pursuing a gainful activity. It serves to determine your tax domicile; pursuing a professional activity in France may constitute one of the criteria taken into account, unless that activity is ancillary there.
A rule that determines tax domicile
The rule concerning tax domicile does not constitute an authorization or prohibition to pursue a gainful activity. It serves to determine your tax domicile based on personal, professional and economic criteria.
Pursuing a salaried or self-employed professional activity in France may constitute one of these criteria. This criterion does not apply when your activity is carried out there on an ancillary basis.
An assessment for each family member
The determination of tax domicile and tax residence is carried out for each member of the household. You may therefore be considered a tax resident of France while your spouse or partner is considered a non-resident.
Professional activity as a criterion
If you pursue several activities or the same activity in several countries, the main activity is taken into account. This is notably the activity to which you devote the most actual time or, when this criterion does not make it possible to decide, the activity that provides you with the largest share of your income.
The role of tax treaties
An international tax treaty may lead to a different qualification from that resulting from French domestic law. When it applies, its rules take precedence in determining your tax residence.