No. The way the Canadian law works, if you are born to a non-Canadian mother in Canada you have dual citizenship. However, Cruz's parents thought they had to positively affirm the desire to take the Canadian citizenship, and since they did not do that, they thought Ted only had US citizenship. A Dallas newspaper researching a piece about him in 2013 found out that the Canadian citizenship actually had been automatic which made him a dual citizen. So he went ahead and renounced it which became official sometime in 2014.
I believe when Cruz was born, Canada is not have dual citizenship. Born in Canada, Canadian citizen. What his parents had to do and did not apparently, was positively affirm he was an American citizen.
The above is just not true. I don't know where some of these ideas came from.
At the time when Ted Cruz was born there was no dual citizenship in Canada (Immigration and Nationality Act of 1952 was the law in force) and the Cruz parents did indeed need to chose. To receive a Canadian BC for Ted they had to chose Canadian citizenship for him. On 15 February 1977, a new "Citizenship Act" came into force. From that date, multiple citizenship became legal but it was not retroactive.