In our project, test procedures and expected test results (test specifications)
are created in a document.
Then, we perform testing on a built product / release.
Here, no test codes nor test tools are involved.
Is this acceptable for unit / integration testing?
What you are doing is “manual testing”.
Manual testing per definition is not, and can never be unit testing.
Manual testing can be used for integration testing, and in fact should be used to some degree, because automated tests cannot spot all forms of unexpected error conditions. Especially bugs having to do with layout and things not “looking right” (which happen to be rather common in web apps).
However, if you have no automated tests at all, it means that your application is insufficiently tested, period. Because it’s completely infeasible to manually test every detailed aspect of an application for every release – no organization would be willing or able to pay for the effort that would require.