No, it is not required to carry the confirmation email or a print-out.
The automatic license plate scanning will let you pass through the barriers without a physical proof of purchase, once the correct license plate has been registered in the Øresund Bridge tolling system.
Please note that it can make sense to have a print out or a picture of the registration for the case of technical issues with the bridge's cameras, though they appear very rarely.
If the barriers do not open you are required to push the “Help” button and will be automatically connected to a bridge worker. If you have a valid toll pass you can show it and they will manually lift the barrier for you.
Related articles:
How is the Øresund Bridge e-vignette checked?
Do I get a confirmation email for my Øresund Bridge e-vignette order?
Are commercial invoices provided for purchases of the Øresund Bridge online ticket?
I bought the Øresund Bridge e-vignette but didn’t get any email, what now? (Theoresundbridge.com)