i think you're pondering about this situation way too deeply.
what you should have done was when you two had eye contact you could waved or gave a hand signal for her to come over to you and if she was interested she would have then. good thing that you didnt yell 20 yards away, she wouldnt have been able to hear you, big dlv. when you knew that she was sunbathing with her friends you should have went over there and opened them up. btw if they were to ask you to put sun cream on them, dont eagerly agree to it, you could say "How much are you willing to pay me?", in a cheeky way.
good ways to open indirectly. assess the situation. its a beach, so get their opinions about some stuff there
"Hey guys, need to get your opinion on something. which is the best place around here to get some ice cream? the reason im asking is that im planning to take my niece to get some ice cream later with her friends later, and she's extremely fussy when it comes to ice cream and she really screams her head off if she doesnt get a good one. know any good places?" this is a good opener because girls always have trouble deciding which places to go and hang out, then you could transition away from the opener.
you could ask them about which is the best surf board wax. if there is a carnival near by, ask them which is the best ride to take you niece to etc.
also you can go direct e.g. "Hey i saw you from over there and i had to tell you that you're just too cute and i wanted to come over to say hi".
hope that i have helped. happy gaming dude
