Read the datasheet.
The LDO you selected doesn't have a constant current mode like LM317, so the fact that it only drops 0.3V@1A isn't significant. It regulates voltage, not current.

It is no doubt that reading the datasheet can find out whether the regulator can limit the current like LM317. The problem is how? This is what I am asking in fact. By looking at the block diagram? Or looking at other specification? If so how?

Thanks again.

 

If you want a lower dropout than that, you may have to go to an op amp/transistor type of constant-current circuit which can reduce the dropout to well less than a volt.

Is there any op amp/transistor type constant-current circuit that can regulate the high side current? I need the load directly connected to the GND due to its nature. Thanks

 

You can do it with a PNP transistor, but sourcing 1A from a 3.7V source will be problematic. You'll have the collector emitter voltage and the voltage drop across the current set resistor to accommodate.

 

It is no doubt that reading the datasheet can find out whether the regulator can limit the current like LM317. The problem is how? This is what I am asking in fact. By looking at the block diagram? Or looking at other specification? If so how?

You can look at the regulator circuitry to see how the voltage is set. You can read the typical applications to see if it lists current source. You can look at the app section, if it exists, to see if any are for current sources. If you take a close look at the LM317, you'll notice that the regulator has no ground pin.

The bottom line is that you don't need a voltage regulator, you need a current source. That can be done with a transistor and a few other components. An opamp, even a low voltage one, is an unnecessary complication with the voltage budget you have.

 

You can look at the regulator circuitry to see how the voltage is set. You can read the typical applications to see if it lists current source. You can look at the app section, if it exists, to see if any are for current sources. If you take a close look at the LM317, you'll notice that the regulator has no ground pin.

The bottom line is that you don't need a voltage regulator, you need a current source. That can be done with a transistor and a few other components. An opamp, even a low voltage one, is an unnecessary complication with the voltage budget you have.

Thanks for your suggestion. I am currently lookng at this op amp transistor current source

http://www.eecs.tufts.edu/~dsculley/tutorial/opamps/opamps7.html
While "Current Out = (V+ - Vref) / R2 "
That mean the Current Out will decrease with battery voltage. How can it be solved?

 

I'd toss the opamp (no value added) and use a zener or a couple diodes in series to set current. But you still don't have much voltage to work with.