The European Commission can bring proceedings against a Member State in the Court of Justice who is able to impose any judgment it considers appropriate (though generally not beyond the request of the Commission).
The European Commission's enforcement role
As an executive body, one of the European Commission's roles is enforcing Union law. From the Commission website:
The Commission is responsible for monitoring whether EU laws are applied correctly and on time. In this role, the Commission is referred to as the 'guardian of the treaties'.
The Commission will take steps if an EU country does not fully incorporate a directive into its national law by the set deadline or has not applied EU law correctly.
If national authorities fail to implement EU laws, the Commission may start formal infringement proceedings against the country in question.
While it is true that administration of EU elections are for the most part the responsibility of each Member State, there are still EU law concerns. Specifically relevant to this question is Article 22(2) of the Treaty on the Functioning of the European Union (TFEU) which specifies that:
Without prejudice to Article 223(1) and to the provisions adopted for its implementation, every citizen of the Union residing in a Member State of which he is not a national shall have the right to vote and to stand as a candidate in elections to the European Parliament in the Member State in which he resides, under the same conditions as nationals of that State. [...]
The Commission thus has a basis for commencing infringement proceedings (alternatively, there's Directive 93/109/EC which provides further details on voting rights and probably provides other bases for infringement proceedings).
Treaty basis for the infringement procedure
In the case of a Member State violating EU law, the Commission may follow the process in TFEU Article 258:
If the Commission considers that a Member State has failed to fulfil an obligation under the Treaties, it shall deliver a reasoned opinion on the matter after giving the State concerned the opportunity to submit its observations.
If the State concerned does not comply with the opinion within the period laid down by the Commission, the latter may bring the matter before the Court of Justice of the European Union.
Then, by TFEU Article 260(1):
If the Court of Justice of the European Union finds that a Member State has failed to fulfil an obligation under the Treaties, the State shall be required to take the necessary measures to comply with the judgment of the Court.
There are no specified limits on what that judgment might be, but it would be presumably within the scope of what the Commission opinion was pursuant to Article 2581. Should the Member State continue failing to comply, Article 260 goes on to specify that fines may be imposed.
In the case of election irregularities
In less severe cases, a declaration of infringement, or a fine is most likely.
However, let's imagine an extreme case where irregularities are so severe, that the election results are essentially meaningless. I really can't see how at minimum the result wouldn't be invalidated. One can reason that since the election did not follow EU rules, it was not a valid EU Parliament election.
However, I don't really want to speculate further as this is quite uncharted territory, and the court will likely have to pull in multiple sources of EU law, and make quite a few new treaty interpretations to resolve such a situation.
Current state of UK proceedings
On the specific UK election irregularities which prompted this question, last month the Commission published a letter which contains the following:
Certain of the problems observed appear to have been recurrences of issues which had arisen previously during the 2014 elections, following which the UK authorities have informed the Commission of their intention to remedy this situation in time for 2019 European elections.
The Commission has written to the UK authorities to ask for further information, as well as an explanation for the apparent lack of measures taken since 2014 to remedy the deficiencies that were previously identified.
The Commission will not hesitate to act in its role of the Guardian of the Treaties should it be required.
It's about as vague as one might expect for a politically sensitive topic, but it sounds like the Commission is currently in or exploring the first paragraph of Article 258 cited above (though if Brexit happens, all this is likely moot).
- I don't have a great source for this, other than checking some cases. That, and the Commission is the plaintiff in a case such as this, and courts don't usually go above and beyond what the plaintiff is asking for.