No, swallowing your own saliva is not haram.
Understanding the Misconception
There's a common misconception that swallowing saliva breaks religious fasts or is otherwise forbidden. This is incorrect. As stated in a reliable source, "This misconception has no basis at all," swallowing saliva is a natural bodily function and perfectly permissible.1
The Islamic Perspective
Islamic teachings emphasize the importance of following natural bodily processes. Swallowing saliva, being a natural and involuntary act, falls under this umbrella. It is not considered an action that would break a fast or be religiously prohibited.
Practical Implications
This understanding eliminates unnecessary worry and allows for a more comfortable and natural observance of religious practices. There's no need to actively avoid swallowing saliva during fasting periods or any other religious context.
1 14-May-2018 (Source provided but lacks a full citation - it needs a source name and publication to be properly referenced)