It deleted only one percent of its flagged illegal content - the same as previous year - and none of it was removed within 24 hours.
The individual could personally be fined up to €5m (£4.4m) if the company doesn't meet the standards.
Heiko Maas said the voluntary efforts of social networks to tackle the problem had not gone far enough.
"This (draft law) sets out binding standards for the way operators of social networks deal with complaints and obliges them to delete criminal content", Justice Minister Heiko Maas said in a statement announcing the plans. Illegal content will have to be deleted within 24 hours, while other offensive content must be taken down or blocked within seven days.
A study published on Tuesday found that Facebook and Twitter are falling short of Germany's benchmark that companies should remove 70 percent of hate speech within 24 hours of discovering the content.
"It is now clear that we must increase the pressure on social networks".
"This isn't sufficient yet", said Mr Maas.
The justice ministry said Facebook takes down only 39 per cent of content reported by users, and only 33 per cent within 24 hours of the complaint, while on Twitter one in 100 reported messages are erased. "We need legal regulations to make companies even more obligated to eradicate criminal offences".
"There can be just as little space in the social networks as on the street for criminal muttering and slander".
While Facebook says it takes its responsibility to fight hate speech and fake news "very seriously", countermeasures shouldn't be applied with a broad brush because that might have an undesirable impact on free speech, Eva-Maria Kirschsieper, Facebook's chief lobbyist in Germany, said at a conference in Berlin in January. Facebook reportedly said its own research showed it had higher rates of removal. This could include libel, hate crime, and threats.
A spokesperson for Twitter declined to comment, but referenced recent changes the company had made to crack down on abusive accounts and make it easier for users to control the type of content that they see.
Online debate around a series of Isis-inspired terror attacks in Europe and mass sex attacks in Cologne on New Year's Eve 2016 has drawn national attention to the issue, as Germany approaches its elections in September.