English grammar: is it possible to automatically verify correctness

Let me try to add another yes/no answer from a slightly different perspective.

1. No, you cannot identify the 'grammatical' correctness of any given sentence using the computational methods of today. Even if we leave aside the fuzzy boundaries of what constitutes a grammatical sentence and even if we ignore the fact that inter-rater agreement with human raters is rather low. Any algorithm available today will misidentify or miss far too many issues that would be immediately obvious to a trained native speaker or native-like editor. The key issue is that a human reader also understands the meaning and can gather the intentions of the writer. They also do not operate with a formal notion of grammar but a rather more holistic approach that will take into account genre, personal style, quotation, etc.

2. Yes, for relatively straightforward texts, it is possible to identify a relatively small subtext of common grammatical issues with some level of reliability. The most common of these include verbal agreement (boys walks), disambiguation of some homonyms (they're vs. there), and common stylistic preferences (overuse of passive). All of these can be very helpful to many struggling writers as well as catch results of copy and paste errors. However, you cannot really speak about 'identifying correctness' in a general sense. Some of the tools available are MS Word grammar checker, After the Deadline, Reverso and they may even offer an API but constructing a usable algorithm like that from scratch would be very difficult (as in: if you have to ask the question, it's probably too difficult for you).