Shortly after the release of the June Public Update, we received notification of a potential issue affecting a subset of Office 2013 Click-to-Run users. In some cases, users running Office 2013 may not be able to launch Office products after the June Public updates are installed. The Office team is aware of this issue and is working on a solution.

Although we have seen this impact less than 1% of our user base, we consider this a high priority issue. For users who have encountered this, the following Microsoft Answers post (I cannot open any office programs after installing updates. - Microsoft Community) describes steps to resolve the issue.
Source: June Public Update issue affecting Click-to-Run users - Office Updates - Site Home - TechNet Blogs