I disagree with a couple of details in Liesmith's otherwise excellent answer. The core point I disagree with is that people can:
... overcome their physical limitations through strength of will.
People can't overcome their limitations. They can only overcome their perceived limitations. This is the key to the movie. People are being told left, right, and center that they can't do XYZ because of their perceived genetic limitations. Vincent achieves success because he drives himself to excel despite the limitations everyone tries to impose on him.
While Vincent is running on the treadmill Anton asks Director Josef if it is possible to exceed your potential, to which Director Josef responds:
No one exceeds their potential. If they did, it would mean we did not accurately gauge their potential in the first place.
This is a very true statement. This society wasn't gauging people's potentially accurately, because spirit/will can be the deciding factor despite a clear genetic/hereditary advantage. This is acknowledged obliquely by Jerome when he states he came in second in the swimming competition even though:
Jerome Morrow was never meant to be one step down on the podium.
So how does any of this relate to your question?
At the meta level, the swimming competition was about overcoming society's expectations, and the fear of not being able to accomplish your goal in the face of the odds. Vincent can only succeed at Gattaca if he overcame his fear of failure, and ignored society's expectations.
At the stated level the swimming competition was a game of chicken. Chicken is all about fear management, and confidence in your ability to handle the outcome regardless of what happens1.
If you watch again there are three key swimming scenes.
- Everything goes as society assumes, and Vincent fails.
- Vincent out swims Anton proving to himself that he can defy society's expectations.
- This is the scene you're talking about.
During the lead up to the third swim Vincent is actively taunting Anton, and this continues while they're swimming. Anton continually expresses fear, uncertainty, and doubt during every bit of dialog while they're swimming. He starts the dialog by saying.
Vincent where's the shore? We're too far out.
Vincent responds by taunting Anton, so they swim on. Next Anton asks Vincent "how can you do any of this?" and "We have to go back." Vincent replies:
We're closer to the other side
He is really saying: "Are you feeling tired because I can go forever" even though this isn't true. The taunt isn't about the truth. It is about breaking Anton's will2. Incredulous Anton asks:
What other side! Do you want to drown us both?
Then Vincent gives the line you mentioned:
You want to know how I did it? This is how I did it. Anton, I never saved anything for the swim back.
This is both a taunt and the truth. The taunt is confirming the implied willingness to drown them both that Anton voiced. The truth is Vincent eschews the fear of failure, and is willing to sacrifice everything to prove society wrong.
And finally, he did in fact have the strength to save his brother otherwise they both would have drowned. Anton needed to be rescued because:
- Anton wasn't in as good of shape as Vincent, and Anton was actually the one who didn't save anything for the trip back.
- Anton gave up mentally so he couldn't access/use his remaining strength3.
Either way like it was mentioned in Liesmith's answer and in the comments below his answer, Vincent wasn't saying "I don't have the strength to swim back."
1: Note: these are the same skills that Vincent needed to succeed as a "borrowed ladder."
2: Before they get hypothermic and/or both don't have the strength to swim back.
3: This is what happened IMO when they were boys and Vincent got tangled in the seaweed. For symmetry reasons I like this as the reason why Vincent needed to save Anton both times.