House Republicans need to focus on turning the administration's retreat on Obamacare into a rout. In light of the administration's announced delay of the employer mandate, they could move immediately to delay the individual mandate as well, and/or the legislation as a whole.
This would put huge pressure on Democrats to explain why big business gets a break but not workers and individuals. It would also highlight how bad Obamacare is—you don't delay something that's a boon to the country. The House GOP could also call Health Secretary Kathleen Sebelius and others to hearings to explain what legal authority the administration has to selectively delay or enforce the legislation. They could also hold hearings to highlight how dangerous the implications of particular parts of Obamacare are both for our health care and our liberty (e.g., the HHS/IRS religious mandate).
In sum, House Republicans could make July the cruelest month for Obamacare, if they seize this moment of opportunity.
Meanwhile, the focus on Obamacare makes it even easier for the House GOP to stay away from immigration. Advocates of the Senate bill are desperate that the House pass something: "'Just vote on something,' urged Janet Murguia, president of the National Council of La Raza. 'We know that if there is a bill that is voted on in the House of Representatives, it will be conferenced with the Senate bill. So just vote on something and let them conference. The conference will be a place where we can negotiate the differences.'" This confirms the fears many of us have had about providing any excuse for going to conference. But the good news is, there's no real-world urgency for the House to pass anything on immigration—since the Democrats won't accept the border-first type legislation the House would pass in any case. So do nothing on immigration. But there is real-world urgency to address the disaster of Obamacare—even the administration has acknowledged that!
So the broader opportunity is for House Republicans to spend the summer discrediting and delaying Obamacare, while allowing whatever minor pressure there was to move on immigration in the aftermath of the Senate action to fade away.