Update Dec. 2020, 5 years later:
(in limited public beta)
In GitHub Enterprise Cloud, the Audit Log now includes Git events and has a new REST API.
Both are available as a limited public beta.
The new Git events will allow you as an administrator to review activities for users interacting with your Git repositories.
You can view events for git.clone
, git.fetch
, and git.push
.
Additionally, the new REST API provides you with another option to interface with your Audit Log events. During the limited public beta, Git events can only be viewed via the REST API and can be exported.
How can you get access to this limited public beta? To be added to the limited public beta, please contact Sales or Support.
Feb. 2021, still for GHE (GitHub for Enterprise):
The Audit Log now includes events associated with GitHub Actions workflow runs.
This data provides enterprise customers with a greatly expanded data set for security and compliance audits.
New events will be incorporated into the audit log when:
- A workflow run is created, completed, deleted, or re-run
- A workflow job is prepared. Importantly, this job will include the list of secrets that were provided to the runner
- A self-hosted runner's version is updated

These new events are only available to customers on the Enterprise plan. All events are available in the REST API, and all events except for workflow run created, workflow run completed, and workflow job prepared are available in the UI and exports.
Learn more about Audit Log events
2015:
Not yet possible through the GitHub API.
But at least, it is possible to export it (since May, 5th 2015) in either JSON or CSV format.

See "Exporting the audit log".