Chief Peter Crespo, Fire Department spokesman, said he could not comment on litigation.
Rodriguez's relatives called 911 at 10:44 a.m. Oct. 1, 2012, after the 24-year-old woman, 36 to 37 weeks pregnant with her first child, fell inside the home in the 2800 block of North Ninth Street.
Paramedics arrived at 10:49 and found Rodriguez on the floor complaining that she felt weak and could not breathe.
The lawsuit says her family told paramedics Rodriguez had a history of blood clots and asthma, and was getting injections of Lovenox, an anticoagulant, to treat deep-vein clotting and reduce the chance of a fatal pulmonary embolism.
The lawsuit says paramedics entered the house without oxygen equipment and did not give Rodriguez oxygen until they got her outside on a gurney.
The suit characterizes the paramedics as callous and unprofessional, adding that they yelled at Rodriguez, "Do your part," and, "You need to sit up."
At the news conference, lawyers distributed a CD that included video from security cameras mounted outside the home that showed the paramedics' arrival and departure.
The video shows the paramedics going into the house with a collapsible "stairchair." Minutes later, the paramedics can be seen struggling to get a barely conscious Rodriguez onto a gurney. The paramedics then belt the heavy woman's torso, head, and arms, and wheel the gurney to the rear of the ambulance. The video also shows that it took both paramedics and the driver to hoist the gurney onto the deck of the ambulance.
The suit seeks unspecified compensatory and punitive damages.
In addition to the city and Fire Department, the lawsuit names as defendants the paramedics who responded to the call, April Smallwood and Lisa McCall.
Joe Schulle, president of Local 22 of the firefighters' union, released a statement calling the paramedics "both excellent medics."
"I would most certainly entrust the care of my family and friends to these two individuals," he said.
Schulle said city paramedics have been "severely overworked" for years.
The lawsuit contends that after Medic Unit 22 arrived at the hospital, its door locks jammed, and it took an extra three to four minutes to get Rodriguez out of the ambulance.
During that time, the suit says, the mother was beyond saving, and the unborn infant - delivered by Caesarean section as his mother was dying - sustained devastating brain damage.
A city investigation found two of 75 emergency vehicles with the same lock problems as Medic Unit 22, the suit alleges.
Furthermore, Smallwood and McCall were suspended for 48 hours without pay in connection with the incident in January 2013 for "conduct unbecoming" a paramedic and violating medical protocols, the suit alleges.
Royce W. Smith, the family's lawyer, told reporters that city investigators called the incident a "catastrophic failure."
"We're bringing this lawsuit for anyone who dares to call 911 and expect a confident, prompt response," Smith said.
Also sued were Trimark Corp. of New Hampton, Iowa; VCI Emergency Vehicle Specialists of Berlin, Camden County; and Horton Emergency Vehicles Co. of Grove City, Ohio.
The suit says the companies are responsible for designing, building, and maintaining city emergency vehicles including Medic Unit 22.