At its peak, a record 8.7Tbps of live video was streamed, according to content delivery network Akamai Technologies.
The previous record of 7.5Tbps was set during US election day coverage on 8 November 2016.
The figures reflect the growing popularity of online streaming.
In the US, 31 million people watched Mr Trump's inauguration on television, compared with 38 million who watched Barack Obama becoming president in 2009.
However, video traffic of Mr Obama's inauguration reached a peak of just 1.1Tbps the same year.