There is no public count of Shakespeare's most-read book, and the question has a catch: he wrote plays for the stage, sonnets, and narrative poems, and his work today circulates in separate editions and in volumes of complete works. The sources consulted do not provide sales or reading data.
What exists are signs of popularity. English Wikipedia says that Romeo and Juliet was among Shakespeare's most popular plays even during his lifetime and that, along with Hamlet, it is one of the most staged. About Hamlet, it says it is one of the most quoted works in the English language and that it had two quarto editions, in 1603 and 1604, while the author was still alive.
Counts of theatrical productions give different results depending on the criterion. One index, which gathers productions announced since 2012 by companies from several countries, a sample rather than a census, puts A Midsummer Night's Dream first, followed by Hamlet, Romeo and Juliet, and Macbeth. Another count, based on reviews from a British theater website, which the author himself advises reading with caution, puts Macbeth first with 141 reviews, then Romeo and Juliet (124), A Midsummer Night's Dream (107), and Hamlet (100).
In short: Hamlet, Romeo and Juliet, and Macbeth always appear among the first, and the order changes with each criterion. Naming just one as the most read is a guess.
On Clavis: Hamlet, Romeo and Juliet, Macbeth.