Can I use GIT directly within Oracle
Really what you're asking for is how can I deploy code/schema/data to a Oracle DB in a remote server, from a Git repo.  This can also be known as orchestration if done as part of a bigger process, or automated code deployment which leads to Continuous Integration/Deployment.

There are there are tons of solutions from your own homebrew options such as scripting it out.  The more commonly used tools for this are Jenkins, and you use .Net, Octopus is great.  Easiest solution is to write a Python/Powershell/whatever script to pull down the latest build, validate it, and execute it as statements then evaluate the return.

These tools work similarly together.  They often provide a web UI for you to communicate with multiple machines across domains and trusts with agents and users executing remotely or directly on the machine as the user. 

For example, Octopus which works for .Net, deploys an agent known as a tentacle on the machine that all actions are executed under and under it's user on the remote machines (Oracle DB Server).  You can then have actions such as 'git pull' directly to your prod ops folder, then the next step verifies it and passes in variables directly from Octopus and continues executing statements.  

Moving fwd, you could use these  as a central part of moving to the SDLC best practices involving C.I.   Jenkins could be used in similar methods for other software.  It is used extensively in management software such as Cloudera Manager to manage big Hadoop clusters.