Well, yes, but you will essentially be building your own MCA. You'll need to amplify and Gaussian-shape the pulses from your photodiode, then use a peak detect & hold circuit followed by an ADC triggered by the peak detector. Your Arduino can then bin the pulse heights into a histogram. If you have high activity source you will need to do something about pulse pileup (overlapping shaped pulses looking like one big pulse, which will distort your spectrum). From memory Atmel microcontrollers usually have internal 8 or 10 bit ADCs built in, so you won't get an incredibly high resolution, but should be enough for basic spectroscopy with a scintillator.

The spectroscopic amplifier is where you need to be really careful to minimise noise in your design. Your shaping time needs to be carefully chosen to match the detector's capacitance (there are lots of papers which talk about this, including application notes from various detector and instrumentation vendors) in order to minimise noise. Spieler's lecture covers this pretty well (look in particular around page 40):

http://www-group.slac.stanford.edu/sluo/lectures/detector_lecture_files/detectorlectures_7.pdf

Well, yes, but you will essentially be building your own MCA. You'll need to amplify and Gaussian-shape the pulses from your photodiode, then use a peak detect & hold circuit followed by an ADC triggered by the peak detector. Your Arduino can then bin the pulse heights into a histogram. If you have high activity source you will need to do something about pulse pileup (overlapping shaped pulses looking like one big pulse, which will distort your spectrum). From memory Atmel microcontrollers usually have internal 8 or 10 bit ADCs built in, so you won't get an incredibly high resolution, but should be enough for basic spectroscopy with a scintillator.

The spectroscopic amplifier is where you need to be really careful to minimise noise in your design. Your shaping time needs to be carefully chosen to match the detector's capacitance (there are lots of papers which talk about this, including application notes from various detector and instrumentation vendors) in order to minimise noise. Spieler's lecture covers this pretty well (look in particular around page 40):

http://www-group.slac.stanford.edu/sluo/lectures/detector_lecture_files/detectorlectures_7.pdf

Dear, you should use a DSP (digital signal processor). DSPs are used for digital signal processing units as they have MAC (multiplication and accumulation), allowing to perform convolution and filtering of signals in real time. Internally work in 16, 24 and 32 bits in arithmetic fracional fixed or floating point. It has 12bits ADC, DAC, GPIO, SCI, SPI, CAN and PWM (linked to the ADC and timers). Look at http://www.freescale.com/ eg DSC (Digital Signal Controllers) (BASIC: MC56F8037EVM: DSC Evaluation Board) or 32-bit Kinetis Microcontroller: FRDM-K64F with cortex M4 has DSP).

provided you have an amplifier, it seems like it may be manageable with an Arduino. As indicated by Germán, however, you may be able to get much better performance using a system more tuned to the problem at hand. There are some efforts going on at the moment to try and exploit the $500 "Red Pitaya" for this purpose, with reported rates of 4 kCps: http://forum.redpitaya.com/viewtopic.php?f=7&t=355

While the Arduino development interface is quite well developed, the Red Pitaya is still a bit in its infancy and so requires some C programming ability to use. 

To ,

Good thought, I was also thinking in the same line as I badly needed an MCA for our graduate lab which is currently running NaI(Tl)s with single channel analyzers. And I don't want to spend too much for MCA.

It came to my mind as I recently tumbled on IUAC, New Delhi's Pheonix Project (the project dates back to 2007) http://www.iuac.res.in/~elab/phoenix/  Phoenix board is also a similar one as Arduino and also runs on Atmel's microprocessors. They developed these experiments http://www.iuac.res.in/~elab/phoenix/experiments/physics.html which can be done with Phoenix. Among them one is radiation detector accessory for Phoenix. The accessory is a set of Preamp, Pulse shaper and amplifier, ADC whose schematics is also available http://www.iuac.res.in/~elab/phoenix/hardware/pics/mca_sch.png  But the sad part is IUAC discontinued Phoenix board and started a new one called ExpEyes http://expeyes.in/ in the same project. In this way the MCA board and GM counter board are currently out of production. But they say they will start the production soon. May be they will make them to be interfaced with ExpEyes as Phoenix board is stopped. But no sorrows, Phoenix project has embraced Free and Open source software and Open hardware principles and hence all schematics are available.

What I was thinking is to build the pulse shaper, ADC etc by using the schematics published by Phoenix (see above link) and make it to interface with Arduino Uno.

Ohh, there are some who tried in a different way with Arduino, watch these youtube videos www.youtube.com/watch?v=6k9pkbhxa00  and  www.youtube.com/watch?v=K5DOR5tAwLk

http://www.iuac.res.in/~elab/phoenix/

http://www.iuac.res.in/~elab/phoenix/experiments/physics.html

http://www.iuac.res.in/~elab/phoenix/hardware/pics/mca_sch.png

http://expeyes.in/

As previously indicated by Brian Pauw, the Red Pitaya board has all the components of a multichannel analyzer. Currently, this board can be bought for about 250 euros: http://store.redpitaya.com/red-pitaya-board.html

More information about the Red Pitaya multichannel analyzer can be found at: http://pavel-demin.github.io/red-pitaya-notes/mcpha

My biggest concern about your idea is the photodiode. If you want to use a scintillator you will need to count photons one by one. That's why people use a PM tube or an MPPC. If you want to use an MPPC you are going to be obtaining nanosecond pulses. A room temperature photodiode will have a dark current larger than your signal.