Supposedly they couldn't. It would be treaty-bound to produce a draft directive or a regulation.

However, it is entirely possible (I'd even say highly likely) that popular initiatives would ask the Commission to legislate on things it's not allowed to by the Treaties.

En un viejo país ineficiente, algo así como España entre dos guerras civiles, poseer una casa y poca hacienda y memoria ninguna. -- Gil de Biedma

Good to have the legal basis spelled out:

Article 11 paragraph 4 Treaty on European Union:

Not less than one million citizens who are nationals of a significant number of Member States may take the initiative of inviting the European Commission, within the framework of its powers, to submit any appropriate proposal on matters where citizens consider that a legal act of the Union is required for the purpose of implementing the Treaties.

The procedures and conditions required for such a citizens' initiative shall be determined in accordance with the first paragraph of Article 24 of the Treaty on the Functioning of the European Union.

(my bold)

Article 24 Treaty on the Functioning of the European Union:

The European Parliament and the Council, acting by means of regulations in accordance with the ordinary legislative procedure, shall adopt the provisions for the procedures and conditions required for a citizens' initiative within the meaning of Article 11 of the Treaty on European Union, including the minimum number of Member States from which such citizens must come.

Little procedural matters have been arranged so far, but we have three limiting requirements (has to be within the framework of the powers of the Commission, has to call for a legal act and has to serve the purpose of implementing the treaties. Concretely that last step means that the legal act has to not only accord to the powers attributed to the EU but also needs to serve the objectives spelled out in the treaties. That step is important for how you'd formulate an initiative. The 'appropriate' qualifier most likely will not go beyond that test.