When Is Picky Too Picky?

I took the girls to their Dr. appointment last week. I usually schedule them both at one time in the summer in an attempt to make it easier and with hopes that there won't be as many germs lurking around the doctor office. Appointments went well, as we expected. We were given the great news once again that we have two healthy, very well growing girls. With Russ and I both being "tall" people this has never been surprising news for us. Our children have been in the 97th% for height since birth.
While talking with the NP she asked about Emily's eating habits. I informed her how picky she is and she was less than impressed. She has been such a challenging child. She doesn't eat meat, ever. She doesn't eat vegetables, ever. And she isn't exactly open to trying new foods. She would literally eat the same meals over and over and over and not appear to get bored. The NP suggested that we buckle down, crack the whip and force her into trying and eating new foods. Which has left me wondering the question...

When is picky too picky?
She is growing, gaining weight, and getting forms of protein in her diet. Do I need to force the issue? Gentle pushing gets no where with her, she is stubborn (I have no idea where that trait came from). The idea of making every meal a battle with a sobbing child seems so unappealing. But is this just me wanting to be comfortable and avoid a needed fight. Or am I avoiding an unnecessary battle by "keeping the peace".
Every where you look you will read conflicting articles and thoughts on this subject. One person would say you are enabling them and giving them control. Them being picky is your fault as a parent. Other say that being picky is a phase that some kids go through. Feed them what they will eat. Two totally different extremes that can both relieve you emotions on the subject and leave you feeling like the worlds worst parent. 
The truth is, as silly as it seems to say my child's eating habits do not define me as a good or bad parent. And some days I need to say that out loud.   As for the fate of Emily. We are still in discussion of what we should do. She comes from a LONG line of many former picky eaters that have all turned out just fine.
Do you have a picky eater?
When is pick too picky for you?
Do you have any advice to share with others?