The call came as police revealed drivers are still failing to comply with the lower limit in Edinburgh, with 57 speeders caught in two hours on the same road last week.
Adventurer Beaumont has joined cycling champions Neil Fachie and Neah Evans in calling for MSPs to pass a bill that would see the limit cut from 30mph in residential and built-up areas.
They told the First Minister: “Anybody who has cycled on the road before will know how intimidating and off-putting it can be when vehicles pass at high speeds, especially for young people and those beginning to build up their confidence.
“Being able to play out and go on bike rides with friends are really important parts of growing up and becoming more independent.
“But the speed of traffic on many of our roads understandably makes parents much more reluctant to afford their children these freedoms.
“Healthy children are more likely to grow up to be healthy adults, and reducing speed limits to 20mph could have a seismic impact on issues like obesity and physical inactivity, which blight the NHS and cost us billions each year.”
Fachie, from Aberdeen, is a Paralympic, world and Commonwealth tandem champion, while Evans, from Cuminestown in Aberdeenshire, is a European team pursuit champion.
They said: “The Scottish Parliament has an amazing opportunity to transform our communities into healthier, greener and more liveable places.”
The bill, introduced by Green MSP Mark Ruskell, is being considered by a Holyrood committee, with MSPs expected to vote on it in June.
Mr Ruskell said: “Elite-level cyclists are backing my bill because they know 20mph speed limits will encourage more people to take everyday journeys by bike, as well as making roads safer for everyone.
“The ball really is in the Scottish Government’s court. It has to choose whether it wants to associate with out-of-step motoring lobbyists or with the overwhelming view from experts, backed up by public opinion, that this is a necessary and life-saving measure.”
However, the Scottish Government’s Transport Scotland agency has yet to give its backing to the move.
A spokesperson said: “We share the view 20mph speed limits are a good idea when implemented in the right environment.
“However, we believe, and it is shared with many local authorities, more evidence and further consideration needs to be given to the impact and consequences of a nationwide default 20mph limit, including an assessment of Scotland’s road network, before the measure proposed in the bill can be fully supported.”
Scottish Conservatives transport spokesman Jamie Greene also had reservations.
He said: “There is no doubt targeted 20mph zones in the right places offer a number of safety and health benefits for all road users.
“What’s right for one town or city may not be right for rural areas or other urban areas, so it is vital whatever measures are introduced to make our roads safer is done with careful consideration of the cost, practicality and effectiveness of the action.”
But Scottish Labour transport spokesman Colin Smyth said: “Whether it’s improving health, road safety or the environment, there is a powerful case for 20mph as the default speed limit in our communities.
“The Scottish Government needs to wake up to the fact the current process to deliver 20mph speed limits just isn’t working with too much inconsistency across Scotland.
“It claims to support reducing speed limits to 20mph, but so far has failed to set out any alternative to the bill to achieve that and it’s time they started listening to the arguments for change.”
Police Scotland said its first major speed camera operation to enforce the capital’s 20mph limit had shown no reduction in speeders.
The 57 drivers detected on Ocean Drive in Leith in two hours on 9 April compared to 105 in two two-hour deployments the previous week.
The road was chosen following community concerns about speeding, with some drivers caught at under 30mph being fined, while some were recorded doing more than 40mph.
Andy Jones, east area safety camera unit manager, described the numbers as “disappointing” considering the deployments were publicised.
Those caught will receive £100 fines and three penalty points on their licence, or be reported to the procurator fiscal for prosecution.
Speed camera vans are being used because fixed cameras are not currently calibrated to detect speeds below 30mph.