But they will still have to complete a pre-departure test before arrival and a PCR test on or before the second day after arrival, but not after eight days.
The need for travellers to self-isolate for ten days after arrival in Scotland is being waived subject to the countries remaining on the amber travel list.
The announcement came hours after a similar decision by the UK Government for England.
The move also covers Switzerland, Iceland, Liechtenstein, Norway, Andorra, Monaco, San Marino and Vatican City.
However, the waiver does not include arrivals from France because of concerns about the prevalence of the Beta Covid-19 variant.
The Scottish Government has not announced the resumption of international cruise ship sailings, unlike in England from Monday, which was part of today’s announcement there.
Transport Secretary Michael Matheson said of the quarantine waiver: “This has only been made possible due to the overwhelming success of our vaccination programme here in Scotland when coupled with successful roll-outs of vaccination schemes in the EU and US.
“Fully-vaccinated travellers will be able to travel to Scotland under this significant relaxation of international travel measures, providing a boost for the tourism sector and wider economy while ensuring public health is protected.
“This new arrangement will be carefully monitored by clinicians and kept under close review as we seek to put Scotland firmly on the path to recovery – but people should continue to think very carefully about travelling – especially given the prevalence and unpredictable nature of variants of concern.”
The Scottish Government said acceptable documents to prove vaccinations were the EU Digital Covid Certificate and the US Centers for Disease Control and Prevention’s white card, known as a CDC card.
However, it still urged Scots not to holiday abroad.
Its spokesperson said: “Our current position remains international travel for holidaying purposes remains risky and subject to sudden change.
"We have said before people should think very carefully about travelling – especially so given the prevalence and unpredictable nature of variants of concern.”
Gordon Dewar, chief executive of Edinburgh Airport, Scotland’s busiest, hailed the quarantine move.
He said: "Confirmation that Scotland will align with England is very welcome and a relief to the industry.
"We are still miles away from normality, but this is a big step in the right direction.
"Safely opening our borders to the EU and USA will kick-start the country's economic recovery which will be beneficial to everyone, especially those businesses and jobs across Scotland and its islands that rely on international visitors.”
However, he also warned: "We must be aware that connectivity will not simply return to where it was in 2019 - it will take time and hard work to do so, and we must put in place things to make that easier.
"We are still in a bizarre situation of people from Scotland having to use a paper certificate that will generate queues when those from England and Europe have access to an app that will allow the use of faster e-gates in Scottish airports.
"The cost of testing in Scotland also remains prohibitive at around twice the cost of our English competitors.”