Ruth Davidson said comments, including homophobic remarks and jibes about her weight, are “completely unacceptable”.
The politician is calling for better protection for young people on social media.
She said people are being “heckled and degraded” online.
She said: “Peoplc call me fat, ugly Tory every day. But every now and then I push back on the homophobic insults because that sort of things is not OK.
“It matters to me to be able to use my position to stand up and say that’s not acceptable.
“We need to understand how ubiquitous social media is for some people. It affects their health and mental health.”
Miss Davidson is calling for tougher criminal guidance on social media communication.
“Transparent rules for engagement and clear penalties - including having accounts suspended - must be outlined so everyone knows where they stand,” she said.
Scotland’s Mental Health Minister Maureen Watt said: “The Scottish Government is committed to making the internet a safer place. We want young people to stay in control and know who to go to if they are at risk.”