Cpl. Nathan Cirillo of Hamilton was gunned down as he and another soldier stood guard at the cenotaph.
"Our family is grieving," a relative reached at the Cirillo home told The Canadian Press.
"Right now is the wrong time to talk."
Military representatives were with the family and the man said he would not be speaking publicly immediately.
Cirillo's mother was reported to have gone to Ottawa while his father and sisters remained in Hamilton.
According to his Facebook profile, Cirillo was an animal lover, outdoor enthusiast and fitness buff.
Photographs show him posing with a dog last month. Other pictures show him with an axe in the woods in uniform, or smiling broadly as he stands in a river in swimming trunks. Another shows a kitten peeking out from his military backpack.
Cirillo, 24, left behind a six-year-old son, according to a friend who asked not to be named. He and the boy's mother had split up, the friend said.
"He loved his son," the friend said. He was the one guy that never picked a fight with anybody. He was just happy go lucky."
The friend called it "messed up" that Cirillo survived a serious motorcycle crash only to be gunned down.
David Cirillo, who identified himself as the soldier's cousin, expressed anger at the killing in a posting on Facebook.
"To the gunman that shot my cousin point blank in front of the Parliament Hill this morning for no reason, you will get what's coming to you," the posting reads.
"You destroyed my whole family for life. If there's a way I can find you I will kill you."
Cirillo, who attended Sherwood Secondary School in Hamilton, was also an online gaming enthusiast.
The soldier was standing on guard at the war memorial near Parliament Hill Wednesday morning when a man opened fire. The gunman was later shot dead in Parliament's Centre Block amid a frantic search for the assailant and any accomplices.
The country's top military commander, Gen. Tom Lawson, issued a statement offering "heartfelt condolences to Cpl. Cirillo’s family, friends and loved ones."
Lawson said Cirillo's death and Monday’s hit-and-run death of a soldier in Saint Jean-sur-Richelieu, Que. are losses "which saddens us all." He added that the military will carry on its missions at home and abroad "in a steadfast, resolute and vigilant manner."
Defence Minister Rob Nicholson also issued a statement to extend his "deepest condolences" and "heartfelt sympathies" to Cirillo's family.
Hamilton Mayor Bob Bratina called it a "such a terrible day" for Canada.
"When you're a soldier you understand that when you go into harm's way that there's always the possibility that the worst could happen, but who would ever think of it standing in ceremony at the cenotaph in Ottawa," Bratina told CP24.
After visiting the family, Bratina said Cirillo joined the cadets as a 13-year-old and became a reservist.
He was only at the cenotaph for a day or so, the mayor said.
"(He) would have been a trained combat soldier sadly taken in this cowardly ambush, which makes the tragedy even greater," said Bratina.
"The sad timing of this man's life to put him there in that position on the day that this occurrence took place is just a horrible irony."
At least two memorial Facebook pages to Cirillo sprang up within hours after the killing.
"RIP Cpl. Nathan," said one comment. "Thank you so much for serving our country. My sincerest condolences to your family. My thoughts and prayers are with you."
Ontario NDP Leader Andrea Horwath, who represents a Hamilton riding, expressed her condolences, saying Cirillo served bravely with the city's Argyll and Sutherland Highlanders — the 91st Canadian Highlanders.
At his home armoury, soldiers watched through the locked gates Wednesday evening as people added flowers and stuffed animals to a growing memorial for their slain comrade.