You need to use a definite article in English any time you wish to specify which of something you mean. If you leave off the, it could mean any two days, which would sound odd with since it debuted because that is also specific. But if you want to indicate that it is those particular two days that have elapsed since the video debuted, you need to use the.
If, on the other hand, you just want to say that this happened during two days at some point in history, you could leave off the (but in that case you should also leave off since it debuted).
The video has been viewed more than 10 million times in two days.
It might have been viewed more than that in total, or it might have been viewed 10 million times then the site was hacked or the server went down and no one has seen it since. It could have taken a long time to become popular over many years, or it could have been an overnight sensation. In this example, we don't know. We only know that during some 48-hour period, there were 10 million views.
In the original, we know that 10 million views occurred on precisely the first two days.
To offer another example of this difference, consider this pair.
In the Catholic faith, a Novena consists of nine days of devotion and prayer.
(Here it is any nine days, at any time of the year.)
During the nine days of a Novena, the faithful will pray the Rosary many times each day, and it is common to fast or abstain from meat, and some people also perform acts of charity.
(Here we are talking about a specific set of nine days, only the exact ones that comprise the act of devotion and not any other.)