The Democratic Governors Assn. has filed a complaint with Ohio election authorities, charging that Republican gubernatorial candidate John Kasich’s recent appearance on Fox News amounted to an in-kind contribution from the network.

The DGA reaped a windfall in counter fund-raising last month when it was disclosed that News Corp., parent company of Fox News, donated $1 million to the Republican Governors Assn. So even if the complaint goes nowhere — it sounds like it’s on somewhat shaky territory of the First Amendment — the publicity doesn’t hurt.

According to its complaint, “There is reason to believe that Fox News Network, L.L.C. violated Ohio election law on August 18, 2010 when John Kasich appeared on a network program, solicited contributions for his campaign and Fox added the graphics “John Kasich (R), KasichforOhio.com” under Mr. Kasich’s image…The Republican Governors Association is actively engaged in working to elect John Kasich in Ohio. It appears that News America, Inc. and Fox News Network, L.L.C. are pursuing a multi-pronged strategy to support and elect Republican candidates.”

Nathan Daschle, executive director of the DGA, said in a statement, “This is fair warning to any Republican governor or gubernatorial candidate who appears on Fox News: We will take legal action when the outfit that is bankrolling your campaign also appears to give you prohibited in-kind contributions.”

The full complaint is here.