Is there an accepted objective criterion of "grammaticality"?

Let me try to provide an alternative answer to @limetom. It is a very good summary of the standard position in formal linguistics but does not answer all aspects of the question.

1. It is possible to have a highly scholarly linguistics that does not dwell on questions of grammaticality at all or only incidentally. Both corpus linguistics and construction grammar are more interested in typical usage than in introspective judgements of speakers about acceptability. Even functionalist approaches are not too concerned with grammaticality as such.

2. The other problem with grammaticality is that there is absolutely no objective measure of it. Chomsky famously copped out and said that any reasonably competent native speaker is just as good a judge of whether a sentence is grammatical or not. That is demonstrably false - one only needs to attend a graduate linguistics seminar to see that people's intuitions differ. In some cases, we can do surveys but even if we get 90% judgement of grammaticality, we still only discovered a variability in the input.

3. That is not to say that we cannot or should not be interested in speaker judgements about language. We want to look at how people correct themselves and each other. We want to see where they seem to be following some pattern in diverging from a standard or simply don't know or commit a slip of the tongue. But we can determine the standard patterns by analysis of actual usage - ie. the "scientific" method of observation. Introspection will still play a role in the analysis but the whole enterprise will not hinge on it.

In short, language is more than sentences (which are an object of writing rather than speech anyway). It is used for communication and self-expression. It is messy and complex. As part of it, people make judgements about acceptability and that is worth studying properly. But grammaticality itself is not a concept you need outside this narrow confine.

You also need grammaticality in some branches of applied linguistics:

For example, when you have an algorithm for generating sentences that is likely to spew out almost anything and you need a formal way of limiting the output of that algorithm. This is where you try to approximate native speaker patterns and turn to their intuitions about these.

Another example, is foreign language instruction. But here you have to construct a sort of inter-language that matches some key patterns of native speech but is not exactly the same. So you turn to generalized grammatical intuitions.

Finally, in contexts like copy editing or essay marking, native speakers turn to their intuitions to identify patterns that are unacceptable for some reason. Accidental slips, divergence from standard into dialect, lack of observance of register, etc. Much more is going on here than 'grammaticality' but often the people performing similar tasks will couch their efforts as avoiding things that are 'ungrammatical'.