WASHINGTON—Many House Republicans are replacing their push to delay or defund the 2010 federal health law with a new strategy: Hang back and see if problems with the rollout continue or get worse.

It is an abrupt reversal of the activist approach of just weeks ago, when Republicans demanded changes to the law in exchange for funding the government or raising the nation's borrowing limit. Now, they say putting the spotlight on technical flaws of the law's health-insurance exchange may be more effective than a direct attack.

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