We propose a method to estimate the household secondary attack rate (hSAR) of COVID-19 in the United Kingdom based on activity on the social media platform X, formerly known as Twitter. Conventional methods of hSAR estimation are resource intensive, requiring regular contact tracing of COVID-19 cases. Our proposed framework provides a complementary method that does not rely on conventional contact tracing or laboratory involvement, including the collection, processing, and analysis of biological samples. We use a text classifier to identify reports of people tweeting about themselves and/or members of their household having COVID-19 infections. A probabilistic analysis is then performed to estimate the hSAR based on the number of self or household, and self and household tweets of COVID-19 infection. The analysis includes adjustments for a reluctance of Twitter users to tweet about household members, and the possibility that the secondary infection was not acquired within the household. Experimental results for the UK, both monthly and weekly, are reported for the period from January 2020 to February 2022. Our results agree with previously reported hSAR estimates, varying with the primary variants of concern, e.g. delta and omicron. The serial interval (SI) is based on the time between the two tweets that indicate a primary and secondary infection. Experimental results, though larger than the consensus, are qualitatively similar. The estimation of hSAR and SI using social media data constitutes a new tool that may help in characterizing, forecasting and managing outbreaks and pandemics in a faster, affordable, and more efficient manner.