* Bigger protest than during Vietnam era, police estimate
* Wisconsin battle has ignited national struggle
MADISON, Wis., March 12 Up to 100,000 people protested at the Wisconsin state Capitol on Saturday against a new law curbing the union rights of public workers that is seen as one of the biggest challenges in decades facing U.S. organized labor.
Madison police spokesman Joel DeSpain estimated the crowd at 85,000 to 100,000 people, which would top the size of protests in Madison during the Vietnam War.
The demonstration, capping three weeks of public protests, came a day after Republican Governor Scott Walker signed into law a bill to eliminate most bargaining rights for many state government workers.
The state Legislature passed the measure this week after Republicans in the state Senate bypassed a Democratic boycott of the chamber.
The battle in Wisconsin has ignited a national struggle over efforts by several budget-strapped state governments to rein in union power.
Republicans say the measures are needed to gain control of deficit-ridden budgets. Democrats and their union backers say Republicans are ramming through union-busting proposals.
The confrontation with unions could be the biggest showdown with labor since President Ronald Reagan fired striking air traffic controllers nearly 30 years ago.
Protesters on Saturday cheered the Democratic state senators who returned to Wisconsin after fleeing to Illinois for three weeks to try to stall the Legislature's consideration of the measure.
"It's so good to be home in Wisconsin," Democratic Senate Minority Leader Mark Miller told demonstrators, who chanted, "Welcome Home" and "We're With You."
"Our fight to protect union rights has become a fight to protect all our rights -- a fight to protect democracy," said Miller. "You have inspired the nation with your passionate and peaceful protests."
In a statement, Republican Senate Majority Leader Scott Fitzgerald, instrumental in shepherding the union restrictions through the Legislature, criticized the Democrats.
"It's an absolute insult to the hundreds of thousands of Wisconsinites who are struggling to find a job, much less one they can run away from and go down to Illinois -- with pay," Fitzgerald said.
Restrictions on public sector unions have been introduced in a number of other U.S. states with Republican governors, including Indiana, Ohio, Iowa, Michigan and Florida. Some Democrats see it as the opening salvo of the 2012 presidential election because unions are the biggest single contributors to the Democratic Party. (Reporting by James B. Kelleher; Editing by Peter Cooney)