What are you using to execute GitVersion? i.e. are you using a script?
GitVersion is not going to update that file for you, but GitVersion will assert the current version number and you can then use something else in your script to update the file.
Personally, almost everything I do in this regard is done using Cake, and there is an addin called MagicChunks, that makes this really easy. For example, this is how I update the package.json file in one of my projects:
https://github.com/gep13/chocolatey-vscode/blob/develop/build.cake#L137-L139
TransformConfig(projectToPackagePackageJson, projectToPackagePackageJson, new TransformationCollection {
{ "version", parameters.Version.SemVersion }
});
But you could equally do something similar in say PowerShell, or bash, or whatever you are using to run the build.
UPDATE:
To do something like this directly in PowerShell, you would need something like this (note, this hasn't been tested):
$gitVersionExe = "./../<Path to where GitVersion.exe lives>";
$output = . $gitVersionExe
$joined = $output -join "`n"
$versionInfo = $joined | ConvertFrom-Json
$version = $versionInfo.LegacySemVer
(Get-Content .\bower.json).replace('1.0.0', $version) | Set-Content .\bower,json
NOTE: Here, within the replace statement, I am making an assumption that you want to replace all instances of a made up version number, 1.0.0, within the bower.json file. In practice, you are going to want to fine tune this replacement, so that you are only replacing/changing, exactly the text that you want to change.
NOTE: You can use any property from GitVersion that you would like, I have just used LegacySemVer as an example