環境

ターミナル: MSYS2(mintty+bash+pacman)
Ruby 2.6 + DevKit
Windows10 64bit

SQLiteのバージョンは問わないという人

SQLite 1.3.13をインストールしよう
コンパイル済だからエラー出ることはないはず。

gem install sqlite3 --version 1.3.13 --platform ruby

mkmf.logを見て

①～package configuration for sqlite3 is not found篇～
②～Cannot create temporary file in C:\篇～
模索中...

出てくるエラー

C:\Users\foobar>gem install sqlite3 --platform ruby
Temporarily enhancing PATH for MSYS/MINGW...
Installing required msys2 packages: mingw-w64-x86_64-sqlite3
警告: mingw-w64-x86_64-sqlite3-x.xx.x-x は最新です -- スキップ
Building native extensions. This could take a while...
ERROR:  Error installing sqlite3:
        ERROR: Failed to build gem native extension.

    current directory: C:/Ruby26-x64/lib/ruby/gems/2.6.0/gems/sqlite3-1.4.0/ext/sqlite3
C:/Ruby26-x64/bin/ruby.exe -r ./siteconfyyyymmdd-foooo-baaaaar.rb extconf.rb
*** extconf.rb failed ***
Could not create Makefile due to some reason, probably lack of necessary
libraries and/or headers.  Check the mkmf.log file for more details.  You may
need configuration options.

Provided configuration options:
        --with-opt-dir
        --without-opt-dir
        --with-opt-include
        --without-opt-include=${opt-dir}/include
        --with-opt-lib
        --without-opt-lib=${opt-dir}/lib
        --with-make-prog
        --without-make-prog
        --srcdir=.
        --curdir
        --ruby=C:/Ruby25-x64/bin/$(RUBY_BASE_NAME)
        --with-sqlcipher
        --without-sqlcipher
        --with-sqlite3-config
        --without-sqlite3-config
        --with-pkg-config
        --without-pkg-config
C:/Ruby26-x64/lib/ruby/2.6.0/mkmf.rb:467:in `try_do': The compiler failed to generate an executable file. (RuntimeError)
You have to install development tools first.
        from C:/Ruby26-x64/lib/ruby/2.6.0/mkmf.rb:552:in `try_link0'
        from C:/Ruby26-x64/lib/ruby/2.6.0/mkmf.rb:570:in `try_link'
        from C:/Ruby26-x64/lib/ruby/2.6.0/mkmf.rb:672:in `try_ldflags'
        from C:/Ruby26-x64/lib/ruby/2.6.0/mkmf.rb:1832:in `pkg_config'
        from extconf.rb:35:in `<main>'

To see why this extension failed to compile, please check the mkmf.log which can be found here:

  C:/Ruby26-x64/lib/ruby/gems/2.6.0/extensions/x64-mingw32/2.6.0/sqlite3-x.x.x/mkmf.log

extconf failed, exit code 1

Gem files will remain installed in C:/Ruby25-x64/lib/ruby/gems/2.5.0/gems/sqlite3-x.x.x for inspection.
Results logged to C:/Ruby26-x64/lib/ruby/gems/2.6.0/extensions/x64-mingw32/2.5.0/sqlite3-x.x.x/gem_make.out