The mechanism being proposed to pass the healthcare bill is called "deem and pass." You can read more about it elsewhere, but it essentially allows the House to "deem" the Senate version to have passed, without taking a vote.
The problem being highlighted is that of Article1, Section 7 in the United States Constitution, which requires that the House and Senate vote on the EXACT same text.
Here are some quotes from a Citizens United challenge to the 2005 Deficit Reduction Act, signed by then President Bush. The authors of this document rightly point out that the exact same bill was not voted on by both the Senate and the House, and was thus unconstitutional.
"That procedure was not followed here.
Rather, on February 1, 2006, the House voted on the engrossed version of S. 1932,
which contained the clerk’s error and, therefore, was not identical to the version of the bill passed by the Senate. See S. 1932, engrossed in Senate (full citation supranote 2); 152 Cong. Rec. H69, H77 (Feb. 1, 2006) (App. 65, 73). The House passed
S. 1932, with the error, by a vote of 216 to 214. 152 Cong. Rec. H68 (App. 64).4
Because the legislation originated in the Senate, the House returned the
legislation to the Senate for transmission to the President for his signature. See 152
Cong. Rec. S443 (Feb. 1, 2006) (message from House to Senate announcing that
House agreed to Senate amendment to S. 1932). When the enrolled bill was prepared,
the Senate clerk changed the provision in section 5101(a)(1) for 36 months of
payment for certain durable medical equipment back to 13 months, as earlier
approved by the Senate. See S. 1932, enrolled in Senate, at § 5101 (App. 44-49).5
The enrolled bill was signed by the Speaker of the House and President pro
tempore of the Senate on February 7, 2006, and transmitted to the President later that
day. App. 113; 152 Cong. Rec. S768 (Feb. 7, 2006). The House, however, had never passed that version of the bill; indeed, the House had never even been sent that
version for consideration. On February 8, 2006, President Bush signed the enrolled bill."
Among the people involved with that noble complaint was Nancy Pelosi, the person attempting to ram the healthcare bill through the House without a vote. This is clearly unconstitutional.