Wa `alaykum As-Salamu wa Rahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
Brother, may you have a happy and blessed Ramadan! Ramadan is a month of discipline, self-control, patience, and good behavior. In Ramadan, Muslims are expected to gain the fruits of Fasting, namely, piety and consciousness of Allah. In this context, Allah, Most High, says in the Glorious Qur'an,(O ye who believe! Fasting is prescribed for you, even as it was prescribed for those before you, that ye may ward off evil)) (Al-Baqarah 2: 183)
As far as Islamic Shari`ah is concerned, there is nothing wrong in swallowing one's own saliva; it does not break one's fast. There is nothing in the sources of Shari`ah to tell that such a thing invalidates one's fasting.
In his response to your question, Sheikh Ahmad Kutty, a senior lecturer and Islamic scholar at the Islamic Institute of Toronto, Ontario, Canada, states,
There is absolutely nothing wrong in swallowing your own saliva; if such a thing had been considered as invalidating fast, then we would have learned about it from the Prophet (peace and blessings be upon him) himself, for it is such an essential matter. We believe the Prophet (peace and blessings be upon him) has communicated to the Ummah in no uncertain terms everything that we must know that is of an essential nature in our religion, especially in matters of worship.
Moreover, Allah says in the Qur'an,(Eat and drink until the white thread of dawn becomes distinct from the black thread of dawn (until the appearance of true dawn).) (Al-Baqarah 2: 187) So we are told to abstain from food and drink while fasting. As Imam Ibn Hazm has rightly pointed out, humankind does not consider things such as swallowing one's saliva, rinsing the mouth or answering the call of nature as akin to eating and drinking.
In conclusion: it is absolutely fine to swallow one's own saliva; it does not break one's fast.
Excerpted, with slight modifications, from: www.islam.ca