Why would he change his focus? Peter the Great was determined to increase the connection between Russia and Western Europe and in order to do that, he needed a Baltic presence; the famed “window to the west.”
To start from the end, the term “window to the west” had been coined by an Italian visiting Russia during the reign of Anna. Attribution it to Peter is a popular anachronism. And, anyway, the normal people are using the doors for going somewhere.
Then, Peter was not focused on Baltic direction until he found that there are no allies to join his further anti-Ottoman adventures (the Great Ottoman War was over and everybody was getting ready for the new major European War). Moving South had clear practical sense because successful elimination of the Crimean Khanate would pacify the Southern border and open for settlement new fertile lands. Further success against the Ottomans would force then to open the Straits for the naval trade (as happened in OTL) simplifying Russian exports from the new Southern territories.
As far as connection with the Western Europe was involved, it already existed, was quite extensive and possession of a part on the Baltic Sea had nothing to do with it. How did Peter get all his "Western" ideas and friends when he was young? All these foreigners lived in Russia either as the merchants or as government's employees.
Sweden was not blocking Russian communication, it was just taking the custom dues for the merchandise passing through its Baltic ports and well after the GNW practically all Russian exports/imports had been carried to the Russian ports by the foreign ships while the Baltic fleet had been happily rotting in Kronstadt. So what exactly would be the difference?
OTOH, emphasis on the Asian markets would provide growing Russian industry with the outlets for their products (as happened in the XIX during the Great Game). As it was, for more than a century Russia stuck with being supplier of the raw materials and buyer of the final products produced by the "West".