Natalie Monson started her food blog 11 years ago and thought the pear logo looked rather nice - although we thought it was a stylised cock and balls at first.
In a legal filing, Apple said the image was too similar to its own logo and would hurt its brand.
Monson says the tech giant is simply "bullying" and she feels a "moral obligation" to fight back.
More than 43,000 people have already signed the petition she and her husband Russ, owners of the Super Healthy Kids website, created last week to try to pressure the company to back down.
"This is a real world example of a small business being destroyed by a giant monopoly because they don't have accountability", Mr Monson told the BBC. "That was so frustrating to us that we thought we had to do something. We can't just be the next victim on the list."
In its filing with the US patent and trademark office, the firm says the Monsons' pear logo "consists of a minimalistic fruit design with a right-angled leaf, which readily calls to mind Apple's famous Apple Logo and creates a similar commercial impression, as shown in the following side-by-side comparison".
It asks regulators to reject the Monsons' trademark application, which the couple first filed in 2017 on behalf of their recipe and meal planning app, Prepear.
Monson said the comparison took them by surprise. The shape of the pear and leaf, which they worked on with a designer, was supposed to allude to the letter "P" in the app's name, she said.
When the Utah-based couple started planning Prepear five years ago, they envisioned the app as a platform where food bloggers could upload recipes, which subscribers could access to plan meals and create shopping lists.
They now have about 21,000 monthly active users, including 3,000 customers who pay the annual $59 fee, according to Monson, an accountant by training. The app, along with the Super Health Kids site, supports five staff, including the couple.
Monson said he initially thought it was a misunderstanding and the two sides would be able to resolve the dispute amicably. The logo has been approved in other countries, including the UK, without dispute and American patent officials hadn't found a problem, he noted.
"We're honestly overwhelmed by how supportive people have been", Mr Monson said. "It's incredible to see how many people share the same frustration that made us decide to take this public. Some of the people reaching out have offered legal and design help, so Mr Monson said he has no plans to back down in the fight anytime soon. "We're going to take it all the way", he said.