The SNP leader admitted the word "national" could be "hugely problematic".
Soldier Dies During Live-Firing Exercise
The man had suffered a serious head wound after being shot during the exercise at the Otterburn training area.
Emergency services were called to the scene at 11.15pm last night, but the soldier was pronounced dead at the scene.
Armed Forces minister Mike Penning said the soldier was from The Royal Regiment of Scotland, and expressed his condolences to the man's family, friends and colleagues.
Mr Penning said: "My thoughts are with the soldier's family, friends and colleagues at this difficult time.
"The safety of our personnel is our absolute priority and, while deaths in training don't happen often, any death is a tragedy.
"As well as a police investigation, MoD accident investigators are looking into the circumstances surrounding this tragic incident.''
Northumbria Police and the Ministry of Defence launched an investigation to establish the exact circumstances surrounding the soldier's death, and a cordon remains in place.
A police watchdog probe was launched after the remains of the 52-year-old were found in a house in Dumfries in February last year.
The accident happened at about 7.50pm on Thursday in Moodiesburn, North Lanarkshire.
More than 70 firefighters were needed to bring the blaze under control at Blochairn Fruit Market in the early hours of Thursday.
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