Can I keep the Mail app on my iPhone from automatically marking messages as read? Is there a way to prevent the iPhone's mail app from marking messages as read automatically when I view them?

Thunderbird on my computer has a way to do this and I find it useful.
There is no setting that will not mark the email, in the Mail app, as read. So, the short answer to your questions is No. 

A workaround that may, or may not, appeal to you, but would involve one more step would be to tap the Edit button in the upper-right corner above the list of email, check mark the ones you want to be kept unread, then tap "Mark", then "Mark as Unread". Again, this option may not work for you, but is probably the fastest way to mark multiple emails as Unread.