The 27-year-old, who is also able to play in the back row, was expecting to be cover for the squad, but was taken by surprise when Edinburgh's Stuart McInally was ruled out with a bulging disk.
"It's not quite sunk in yet," Bryce told the BBC. "It's all happened pretty fast.
"I managed to get a quick phone call into my old man. He was on a building site and he was quite taken aback!
"I haven't spoken to Stuart yet. I feel quite bad the way it's come about through an injury and my thoughts are with him."
Bryce had travelled to Italy for Scotland's warm-up game in Turin as cover, but did not come off the bench.
And with only two caps to his name, Bryce assumed he was travelling to Scottish rugby headquarters for a check-up, with no idea that he might be about to receive the biggest news of his career.
"I just thought [Scotland doctor] James Robson was going to check me over from the Glasgow game as I realised I was going to be back-up cover," he added.
"I thought I'd come in then go straight home.
"When I went to Italy I thought that was it. I was coming for cover, just in case. I genuinely never thought anything like this would happen.
"I have a wee boy - his name's Layton - and I've just booked his birthday party and I'm going to miss it. I need to let him know. His birthday's in October."
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