The men part doesn't really matter because birth rates are determined more by women's behaviors than anything else. And the paper does focus on women.

  Overall, the diffusion of the iPhone explains 33–52% of the decline in
  the general fertility rate among women aged 15–44.

That might only be part of the issue. There has been some "experiments" where people are asked if they'd prefer to give up their phone for some period of time, or go without sex in that same period. Women were more likely to prefer their phones over sex and other studies show that women generally spend more time on their phones than men does.

never