Nadal gave her a kiss on the cheek and apologized. After the match, he gave her his headband before saluting the crowd.
"One of the more scary moments I've had on the tennis court -- the ball was quick, and straight on [her] head," he said. "I was so scared for her. Very happy that you're good. You're very brave, so well done."
Nadal's 6-3, 7-6 (4), 6-1 victory was tough. He'd dropped only 10 games in three previous wins over Delbonis, a left-handed Argentine.
The first two games on Rod Laver Arena took 18 minutes, with Nadal missing six breakpoint chances before Delbonis held for 1-1.
The top-ranked Nadal converted 1 of 9 break-point chances in the first set and 0 of 9 in the second, but converted 2 of 2 in the third.
Medvedev, a US Open finalist, was leading 5-0 in the second set when he called the trainer to help stem the blood from his nose. He said nosebleeds happen to him sometimes, he wasn't bothered too much, and it wasn't related to tension.
"No, no, not at all. If it would be, I would be bleeding every match I play," he said with a smile.
Medvedev said shifting from outdoor conditions to indoors helped him against Martinez.
"Yeah, of course it changes a lot. Even, like, it was gray when we stepped on the court. We just played one game on the, let's say, outdoor court," he said. "As soon as the roof is closed, everything is different. It gets more hot, more humid inside. The ball goes faster. I think the sound of the ball even is different. So everything is completely different.
"Today, I think it was in my advantage. Sometimes can be in your disadvantage."
Medvedev fended off four break-point chances to hold in the sixth game of the third set and finished strongly.
The mercurial Australian has pledged $200 to bushfire relief for every ace he serves this month, and the tally is mounting.
Taylor Fritz's 4-6, 6-7 (5), 7-6 (4), 6-2, 6-2 comeback victory over two-time Grand Slam finalist Kevin Anderson gives the United States five men in the Australian Open's third round for the first time since 2008.
The 29th-seeded Fritz has never been further at a major tournament. This was the second time in his career that he has managed to win a match in five sets after dropping the opening two.
The other trio advanced Wednesday. Querrey and Sandgren meet each other Friday for a berth in the fourth round, while Paul will play Marton Fucsovics of Hungary.
Wawrinka, the 2014 Australian Open champion, defeated Andreas Seppi 4-6, 7-5, 6-3, 3-6, 6-4.
Thiem broke local wild-card entry Alex Bolt's serve in the third game of the fifth set and advanced to the third round with a 6-2, 5-7, 6-7 (5), 6-1, 6-2 win.
Thiem had to deal with a partisan crowd at Melbourne Arena but said he enjoyed the occasion.
"It was a tough match," Thiem said. "Alex played great, with great crowd support. Great fun. It's nice to play with such atmosphere"
No. 7 Alexander Zverev, who has pledged "every cent" of his prize money to the bushfire relief effort if he wins his first major title this year in Australia, advanced at about the same time with a 7-6 (5), 6-4, 7-5 win over Egor Gerasimov.
Zverev, who won the ATP Finals two years ago, has pledged $10,000 for every match he wins at the very least.
Heavy morning rain showers delayed play on the outside courts for at least 90 minutes and for up to three hours.
The rain stopped late morning, and the weather was mostly cloudy with gusting winds. Still, organizers were again forced to postpone matches because of the weather, rescheduling seven women's and six men's first-round doubles matches to Friday.
The Associated Press contributed to this report.