ICU-21450 Migrates ICU4C document build with doxygen from Travis CI to GHA CI.

.github/workflows/icu_ci.yml

@@ -13,6 +13,21 @@ on:
 
 jobs:
 
+  # ICU4C docs build using doxygen..
+  icu4c-docs-build:
+    runs-on: ubuntu-latest
+    steps:
+      - uses: actions/checkout@v2
+
+      - name: ICU4C doc
+        run: |
+          sudo apt-get -y install doxygen;
+          cd icu4c/source;
+          ./runConfigureICU Linux --disable-renaming;
+           # Fail if 'warning:' appears in doxygen's output, but ignore warnings from file Doxyfile.
+           # Regex note: (?! ... ) is a negative lookahead. Succeed if the pattern is not present. 
+          set +o pipefail && make doc 2>&1 | tee doxygen.log && ( ! grep -P 'warning:(?! .* file .?Doxyfile)' doxygen.log )
+
   # ICU4J build and unit test
   icu4j-build-and-test:
     runs-on: ubuntu-latest