Georgia and Gibraltar will be the Republic’s other competitors.
“I’m feeling pretty positive about it. I’m thrilled, delighted and honoured and can’t wait to get on with all the games,” stated the Republic boss.
“I think we could beat anyone at home when we have got the crowd behind us”
Denmark saw off the Republic in their November 2017 World Cup play-off and the sides drew 0-0 in both of their recent Nations League meetings.
The Republic had originally been drawn in Group C together with the Netherlands and Germany but were handed a reprieve since the bands already comprised two of the 12 host countries for Euro 2020.
“Switzerland were one of the best teams in Europe lately – I saw them at the Nations League and that I thought they were outstanding,” added McCarthy.
“Denmark are our nemesis. They knocked us out of this World Cup play-offs and completed ahead of us at our Nations League team as well but that’s now before.
“I’m not saying our team is simpler but the Netherlands and Germany would have been a tough ask.”
The Euro 2020 qualifiers will occur between March and November 2019, together with the winner and runner-up of each class going through directly.