Actually, I think your numbers are way off. First of all, VZW's capex last year was $10.5 billion. And their guidance for 2015 is $18-$19 billion.
Secondly, the $10B figure you quote regarding Vodafone (They've actually increased it to $11.1B, by the way) is over two years. 2015, and 2016. And it is part of a special project that Vodafone is undertaking known as Project Spring, which they are using to actually upgrade their whole network. It came after Vodafone divested itself of its interest in Verizon for $130B. It includes upgrading their 3G network in some major cities, and upgrading their overall 4G network. Additionally, I think you will find that Vodafone's capex is for their global network, and not just Europe.
Verizon, on the other hand already has a very extensive 4G (LTE) network in the US that they upgraded to over three years ago. The simple fact is, that given the size and topographic challenges of the US it is more expensive to build out and maintain a wireless network here. And, as I alluded to before, the return on investment in infrastructure in places like Wyoming, Montana, Idaho, etc. is far less than it more population dense areas, such as the Eastern half of the US, and most all of Europe. It's just simple physics.