Not likely. Let me explain:
Viruses spread by modifying programs. But on iOS, each application has a sandbox, which makes it hard for the infected program to infect other programs.
As for malware, that's hard to run, too. Each application is code-signed by its developer and vetted by Apple. Code signing is encryption that makes it difficult for anyone but an app's original developer/team to see, let alone modify, some app's code.
Imagine that your phone is a community, with houses that only their occupants (apps) have access to. Surrounding the community is a set of tall walls. At its entrance is an armed guard that checks the IDs of each app that moves in, to make sure that the app is allowed to be there.